Headlamps on, both brake lamps are on;hit brake, left brake lamp brightens, right lamp remains dim;headlamps off, hit brake, only left lamp lights. What could be the problem?

Headlamps
You can use the headlamps two different ways.
• Place the headlamp switch in the HEAD position for normal operations
• Place the headlamp switch in the OFF position for automatic lamp control (ALC) operation. During ALC the low beam headlamps are ON for daytime running lamp (DRL) operation in daylight conditions at reduced intensity or low beam headlamps ON at full intensity with exterior park lamps ON in low light conditions.
The body control module (BCM) applies a pulse width modulated (PWM) voltage to the headlamp module and high beam relay in the underhood fuse block. The ground for the headlamp module is supplied from G113 or G115. For normal manual low beam operation when the headlamp switch is place in the HEAD position, ground from G202 is applied through the headlamp switch and to the BCM through the headlamp switch headlamps ON signal circuit. The BCM then applies a PWM voltage to the headlamp module/high beam relay voltage supply circuit. The headlamp module then applies battery positive voltage through the left front (LF) and right front (RF) LO Beam fuses to the left and right low beam headlamps. The ground for the left headlamps is supplied by G101 and the ground for the right headlamp is supplied by G100.
For high beam operation, ground is applied at all times to the headlamp dimmer switch from G202. When the headlamp dimmer switch is placed in the HIGH position, ground is then applied through the headlamp dimmer switch signal circuit to the BCM. The BCM then applies a ground to the headlamp high beam relay control circuit. This energizes the HI Beam relay, the relay then applies battery positive voltage through the LF and RF HI BEAM fuses to the LF and RF high beam headlamps. The ground for the left headlamps is supplied by G101 and the ground for the right headlamp is supplied by G100.
The BCM then sends a class 2 message to the instrument cluster in order to illuminate the high beam indicator lamp. If the headlamp switch is left in the HEAD position, the inadvertent power control feature will turn OFF the headlamps after 10 minutes after you turn the ignition switch to the OFF position. If you place the headlamp switch in the head position after the ignition switch has been turned OFF, or if the ignition switch is in ACCY position, the headlamps will remain ON until you turn them OFF or until the battery runs dead.

It's in relay block 1
No.
Device
Rating
Description
1
Window Relay
--
Power Window
2
Reverse Relay (Allison® MD and UZF)
--
Backup Lights and Backup Alarm (Allison® MD and UZF)
3
Headlamp High Beam Relay
--
Headlamps
4
Lighting Relay
--
Headlamps
5
Headlamp Relay
--
Headlamps
6
Turn Trailer Relay - Left
--
Trailer Left Rear Turn/Stop Lamp
7
Tail Lamp Relay
--
Tail Lamps
8
Marker Lamp Relay (JE4, RQ2 and U86 or JE4 and RQ3)
--
Clearance, Identification (ID) and Marker Lamps
9
Turn Trailer Relay - Right
--
Trailer Right Rear Turn/Stop Lamp


Relay Block 1
In the right top of the I/P, under the access panel

https://www.carcareninja.com/tail-lights-not-working-but-brake-lights-are/#:~:text=The%20common%20causes%20the%20tail%20lights%20do%20not,it%20does%20not%20allow%20the%20circuit%20to%20complete.
This post has a few suggestions that may help you find your problem.

I would check the fuses first. Headlights are typically on a separate fuse then the parking lights. If the fuse is fine check and see if the Bulb is burnt out but it is rare for both lights to go out at the same time.

Your right brake light is fried. The fact that you see rear lights on both sides has nothing to do with the fact that the right brake light is out. Your brake lights and rear running lights are not the same bulbs. Go to Advance Auto Parts. They may even replace it for you. Not a big deal.

Yea, take it to a qualified repair shop ! Could be anything , and with out doing some test couldn't make a suggestions .
Headlights/Daytime Running Lights (DRL) Circuit Description
The instrument cluster controls the headlamp systems that include daytime running lamps (DRL). The system allows the headlamps to operate at a reduced intensity when all of the following conditions apply:
• The engine is running.
• The transaxle is not in the PARK position.
• The normal headlamps are not on.
The instrument cluster turns the DRL system on by grounding CKT 592, thereby energizing the DRL relay. This action allows a reduced voltage of 6 volts to flow across the headlamps. If the vehicle has the optional fog lamps, the fog lamp relay will also be energized. The park lamps will turn on and the fog lamps will operate normally. When the headlamp switch is turned to the ON position, the instrument cluster allows normal headlamp operation.
The instrument cluster also has an input from the park brake. If the park brake is in the ON position, the DRL will remain disabled until the park brake is placed in the OFF position. On vehicles with an automatic transaxle, the instrument cluster will disable the DRL until the transaxle has been taken out of the PARK position.

You have a gremlin ! No , just kidding . No it's not the ignition switch ! The headlamps right an left have there own fuse in the under hood fuse box . LH HDLP - RH HDLP . Power direct to the head lamps an they ground through the headlamp switch when placed in high or low beams . Next time they come on wiggle the headlamp switch an see if they go out ! Does your vehicle have auto lights ? Day time running lamps ? The BCM - body control module controls Automatic lamp control !
Battery voltage is applied to the headlamp bulbs at all times. The headlamps are controlled by supplying the ground for the lamps through the turn/headlamp switch assembly or the body control module (BCM). When the turn/headlamp switch is in the ON position, the ground is supplied through CKT 150.
The BCM controls the daytime running lamps (DRL). The system allows the headlamps to operate at a reduced intensity under the following conditions:
?€¢
The engine is running.

?€¢
The transaxle is not in the PARK position.

?€¢
The normal headlamps are turned OFF.

The BCM activates the system by grounding CKT 592. This action energizes the DRL relay. This condition allows a reduced voltage to flow to the headlamps. When the headlamp switch is turned to ON, the headlamps operate normally.
The parking brake provides an input to the BCM. When the parking brake is engaged, the DRL remain disabled until the parking brake is disengaged.

There are several reasons why your headlamps an running lights don't work , the headlamp switch is a good one ,plus relays an the DIM - dash integrated module an associated wiring ! Here is how the headlamps work , right out of Buick factory service info !
The dash integration module (DIM) provides battery positive voltage to the signal circuits of the headlamp switch. The signal circuits include the headlamp switch headlamps on signal, the flash to pass switch signal and the high beam input. With the headlamp switch in the ON position, a ground path is available for the headlamp switch headlamps on signal circuit of the DIM through the headlamp switch. The DIM provides ground to either the headlamps high beam relay control circuit or the headlamps low beam relay control circuit of the low beam circuit. The position of the headlamp dimmer switch determines which relay control circuit has ground. The DIM supplies battery positive voltage to the relay control circuits, if the headlamps are necessary. The fuse block - underhood supplies battery positive voltage to both relay switch circuits. When the low beam relay control circuit is energized, current flow is to both low beam fuses and to the low beam headlamps. The headlamps have ground at G103 and at G401. When the headlamp high beam relay control circuit is energized, current flow is to both high beam fuses and to the high beam headlamps. The high beam circuit is divided into a left and right side circuit. The current flow of the left high beam is from the fuse to the left high beam lamp. From the lamp, the current flows to the normally-closed contact of the DRL relay to G103. The current flow of the right high beam is from the fuse to the right high beam lamp. From the lamp, the current flow is to ground at G401. When the headlamp dimmer switch lever is pulled toward the driver, the flash-to-pass switch closes. This grounds the DIM flash to pass switch signal circuit. In response to this input, the DIM energizes the HDLP HI BM relay. Both high beams remain on until the driver releases the switch handle. If the low beam headlamps were on during a flash to pass operation, the low beams remain on. The headlights may be switched to high beam with the opposite movement of the headlamp dimmer switch lever.
By running lights , do you mean park lamps or day time running lamps ?
Park, Tail, License and Marker Lamps
The park lamps, including the license lamps, are turned on when the headlamp switch is placed in the PARK or ON position or anytime the headlights are requested. The fuse block - rear supplies battery positive voltage to the switch contacts of the Park LP relay. When the headlamp switch is in PARK or in ON, the park lamp switch on signal circuit of the DIM has a ground path. The DIM provides battery positive voltage to the control circuit of the Park LP relay. When the park lamps and/or the tail lamps are ON, the DIM energizes the park lamp relay control circuit. This closes the switch contacts. Current flows from the Park LP relay to the LH PARK LP and the RH PARK LP fuses. From the fuse, the current flows to the park lamps. The left park lamps are grounded at G103, G401 and G402. The right park lamps and the right tail lamps are grounded at G103 and G401. The DIMR fuse supplies battery positive voltage to the license lamps and to the DIM instrument panel lamp fuse supply voltage - 1 circuit, for interior lamps dimming. The license lamps are grounded at G401.
The best advice I can give is that you take this to a ASE certified professional repair facility unless your a qualified automotive electric repair person !
Diagnostic Starting Point - Lighting Systems
Begin the system diagnosis with the Diagnostic System Check - Lighting Systems . The Diagnostic System Check will provide the following information:
?€¢
The identification of the control modules which command the system

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The ability of the control modules to communicate through the serial data circuit

?€¢
The identification of any stored diagnostic trouble codes (DTCs) and their status

The use of the Diagnostic System Check will identify the correct procedure for diagnosing the system and where the procedure is located.

The

Which insturment cluster does your's have ? electronic instrument cluster or standard instrument cluster !

Standard insturment cluster - speedo / odometer
Speedometer/Odometer
The speedometer and odometer provides speed information based on a speed/distance pulse signal from the vehicle speed sensor (VSS). The VSS is mounted on the speedometer driven gear in the transmission.
Could be the VSS is bad .
Horn The horn system includes the following:

• horn relay
• horn
• clock spring
• steering wheel control switch harness
• steering wheel pad horn switch, which is part of the driver side air bag module. For additional information, refer to Section 501-20B .

The horn is located behind the front bumper mounted to the radiator support.

Is the air bag light lit on the insturment cluster ? If it is the clock spring could be the problem . If not lit check the circuits from the horn relay .

VEHICLE RELAYS Operation Diagnosis .

Which headlamps don't work ? Low beams , high beams , both ?

• The high beams are inoperative

• Bulbs.
• Circuitry.
• Multifunction switch.

• GO to Pinpoint Test C .

• The low beams are inoperative

• Bulbs.
• Circuitry.
• Headlamp switch.
• Multifunction switch.
• LCM.

• GO to Pinpoint Test D .

Principles of Operation
NOTE: Due to varying wattage ratings and resulting current draw, certain aftermarket headlamp bulbs may cause the lighting control module (LCM) to shut down the short circuit protection, resulting in the low beams becoming inoperative. Verify that the headlamp bulbs meet Ford specifications.
Headlamp System The headlamp system consists of:

• two aerodynamic halogen headlamp bulbs and retainers
• headlamp switch
• multifunction switch
• lighting control module (LCM)
• circuit wiring and circuit protection

In normal operation:

• With the multifunction switch in the low beam position, power is supplied by the LCM to the low beam elements through the multifunction switch.
• With the multifunction switch in the high beam position (forward detent), power is supplied to the high beam elements and to the high beam indicator in the instrument cluster.
• When the multifunction switch is pulled toward the steering wheel, the flash-to-pass switch (part of the multifunction switch) is closed and power is supplied to the high beam elements. When the switch is released, they return to low beam operation.
• The battery saver feature of the LCM will automatically turn off the courtesy lamps, demand lamps, headlamps, or parking lamps under the following conditions:
  • The ignition switch is in the OFF position.
  • The headlamp switch is in parking lamps ON or headlamps ON positions.
  • After 10 minutes.

Headlamp Switch The headlamp switch is a multi-position switch.

• The positions include OFF, PARK, and HEAD.
• With the headlamp switch in the PARK position (first detent), a signal is supplied to the LCM which powers the front parking lamps, the tail lamps, the side marker lamps, and the license lamp.
• With the headlamp switch in the HEAD position, power is supplied to the headlamps through the LCM and the multifunction switch.
• The panel dim switch (part of the headlamp switch) is located above the headlamp switch. The panel dim switch operates only with the parking lamps or headlamps on.
• When the panel dim switch is held in the up position, the instrument panel lighting will progressively brighten.
• When the panel dim switch is held in the down position, the instrument panel lighting will progressively dim.

For additional information, refer to Section 413-00 .
Headlamps On Warning Chime If the headlamp switch is left in either the PARK or HEAD position and the driver door is opened, the LCM will sound a headlamps on reminder chime when the following conditions occur:

• The ignition switch is OFF with the ignition key out.
• The driver door is ajar.

For additional information, refer to Section 413-09 .

Inspection and Verification

1. Verify the customer concern by operating the system.

1. Visually inspect for obvious signs of mechanical and electrical damage.

Visual Inspection Chart Mechanical Electrical

• Headlamp switch

• Central junction box (CJB) fuse(s):
  • 6 (15A)
  • 15 (15A)
  • 21 (15A)
  • 23 (15A)
  • 27 (20A)
• Battery junction box (BJB) HI-BEAM HEADLIGHTS (20A) fuse
• Circuitry
• Headlamp bulb
• Lighting control module (LCM)
• Loose or corroded connections

make sure the terminals are clean at the headlamps. check the black wire to ground for open circuit. an open ground will cause a dim headlamp if there is no way to test wire jump the black wire to ground and see if lamp brightens