Turn signal lamp front and rear and indicator stays on and does not blink

No flasher relay , the BCM - body control module does this . Your best bet ,take it to a qualified repair shop.

Turn Signal Lamps
Battery voltage supplied to the body control module (BCM) from the fuses listed below is used for turn signal and hazard lamp operation.
• LT TRN/SIG fuse for the left turn signals
• CTSY/RF TRN fuse for the right front turn signal
• RR TRN/SIG fuse for the right rear turn signal
When the turn signal switch is place in either the LEFT or RIGHT position, ground from G300 is applied through the turn signal switch signal circuit to the BCM indicating the turn signal request. In response to this signal, the BCM applies a pulsating voltage to the front and rear turn signal lamps supply voltage circuits cycling the lamps ON and OFF. At the same time the BCM sends a GMLAN serial data message to the instrument panel cluster (IPC) to cycle the appropriate turn signal indicator ON and OFF.
Ground for the turn signal lamps is supplied as follows:
• G104 provides ground for the left and right front turn signal lamps
• G402 provides ground for the left rear stop/tail/turn signal lamps
• G403 provides ground for the right rear stop/tail/turn signal lamps

Hooking up a factory or someother type scan tool an checking for DTC'S diagnostic trouble codes in the BCM would be the first thing to do.

Turn Signal Lamps
The body control module (BCM) is responsible for controlling the turn signals and hazard lamps. When the turn signal switch is placed in either the left turn or right turn position, the BCM receives a signal from the turn signal switch. The BCM then sends an on-off voltage signal to either the left or right turn signals and their instrument panel cluster (IPC) indicator. When the hazard switch is pressed, the BCM receives a signal from the hazard switch. The BCM then will send an on-off voltage signal to all the turn signal lamps. All the turn signal lamps will flash including both IPC turn signal indicators.
Stop Lamps
The body control module (BCM) is responsible for controlling the stop lamps. When the driver presses the brake pedal, the BCM will receive a signal from the brake pedal position sensor. The BCM then supplies battery positive voltage to both left and right stop lamp assemblies and to the center high mounted stop lamp (CHMSL). The stop lamps are grounded at G401 and G403. The CHMSL is grounded at G302.

Something is messing with the BCM'S mind ,bad input data ,shorted circuit etc... Hooking up a professional type scan tool is needed to read BCM DTC'S - diagnostic trouble codes an view live data to the BCM .

DTC B3948 02: Left Front Turn Signal Circuit Short to Ground
DTC B3948 05: Left Front Turn Signal Circuit Short to Battery or Open
DTC B3949 02: Right Front Turn Signal Circuit Short to Ground
DTC B3949 05: Right Front Turn Signal Circuit Short to Battery or Open

DTC B3950 02: Left Rear Turn Signal Circuit Short to Ground
DTC B3950 05: Left Rear Turn Signal Circuit Short to Battery or Open
DTC B3951 02: Right Rear Turn Signal Circuit Short to Ground
DTC B3951 05: Right Rear Turn Signal Circuit Short to Battery or Open

Your best bet ,take it to a qualified repair shop.

The turn signals are controlled by the BCM
Turn Signal Lamps
The body control module (BCM) is responsible for controlling the turn signals and hazard lamps. When the turn signal switch is placed in either the left turn or right turn position, the BCM receives a signal from the turn signal switch. The BCM then sends an on-off voltage signal to either the left or right turn signals and their instrument panel cluster (IPC) indicator. When the hazard switch is pressed, the BCM receives a signal from the hazard switch. The BCM then will send an on-off voltage signal to all the turn signal lamps. All the turn signal lamps will flash including both IPC turn signal indicators.

I would suspect a problem with the BCM or maybe corrosion in the under hood fuse box . Wires from the BCM to the front turn signals go through the underhood fuse box.
Your best bet ,take it to a qualified repair shop that has a diagnostic scan tool that can read BCM DTC'S - diagnostic trouble codes
DTC B3948 Left Front Turn Signal Circuit
The body control module (BCM) controls the left front turn signal, hazard lamp, and daytime running lamp (DRL). The BCM supplies voltage to control the lighting functions. The BCM monitors the left turn signal lamp supply voltage circuit for proper operation.
DTC B3949 Right Front Turn Signal Circuit

Get the left signal to flash fast then walk around the car to see which light isn't flashing then change the bulb or socket if they all appear to be flashing (front and rear) turn on the right turn signal then look into the front and rear lens and count the number of bulbs flashing, now do the same thing on the left. Many times there's more than 1 bulb in each lamp. I hope this helps. Take care.

DTD ??? DTS
Do you know how to test automotive electrical circuits using a DVOM - digital volt ohmmeter ? Looking at a wiring diagram to see what all is involved in making the right turn signal work ? Free wiring diagrams here http://www.bbbind.com/free-tsb/ Enter vehicle info. Year , make , model an engine size . Under system click on lighting ,then under subsystem click on turn signal lamps . Click the search button then the blue link . Looking at the diagram for the turn signals i see the turn signal switch is a input to the BCM - body control module . The turn signals are controlled by the BCM . Having a professional scan tool to check input data to the BCM an out put control for the turn lamps on the right side would save time .
https://www.youtube.com/watch?v=YaYtCVBodpw&t=98s

Turn Signal Lamps
Battery voltage supplied to the body control module (BCM) from the fuses listed below is used for turn signal and hazard lamp operation.
• LT TRN/SIG fuse for the left turn signals
• CTSY/RF TRN fuse for the right front turn signal
• RR TRN/SIG fuse for the right rear turn signal
When the turn signal switch is place in either the LEFT or RIGHT position, ground from G300 is applied through the turn signal switch signal circuit to the BCM indicating the turn signal request. In response to this signal, the BCM applies a pulsating voltage to the front and rear turn signal lamps supply voltage circuits cycling the lamps ON and OFF. At the same time the BCM sends a GMLAN serial data message to the instrument panel cluster (IPC) to cycle the appropriate turn signal indicator ON and OFF.
Ground for the turn signal lamps is supplied as follows:
• G104 provides ground for the left and right front turn signal lamps
• G402 provides ground for the left rear stop/tail/turn signal lamps
• G403 provides ground for the right rear stop/tail/turn signal lamps
• The driver door module (DDM) provides a low reference for the left outside rearview mirror turn signal
• The front passenger door module (FPDM) provides a low reference for the right outside rearview mirror turn signal

Stop Lamps
The brake pedal position sensor is used to sense the action of the driver application of the brake pedal. The body control module (BCM) provides a 5-volt reference voltage and low reference to the brake pedal position sensor. The position sensor provides an analog voltage signal back to the BCM that will increase as the brake pedal is applied. When the brake is applied, the BCM will then apply battery voltage to the left and right stop/turn signal lamp supply voltage circuits and the center high mounted stop lamp (CHMSL) supply voltage circuit illuminating the stop lamps. At the same time the BCM applies battery voltage through the signal circuit to the transmission control module (TCM) and the engine control module (ECM) indicating that the brakes are applied.
Ground for the stop lamps is supplied as follows:
• G402 provides ground for the left rear stop/tail/turn signal lamps
• G403 provides ground for the right rear stop/tail/turn signal lamps
• G301 provides ground for the CHMSL
The stop lamps on this vehicle will not illuminate unless the ignition is in the accessory, run, or crank positions. When the ignition is in the OFF position the stop lamps will not illuminate when the brake pedal is applied.
Battery voltage from the LT TRN/SIG and RT TRN/SIG fuses located in the rear fuse block is also used by the body control module (BCM) for stop lamp operation.

Turn Signal Lamps
Battery voltage supplied to the body control module (BCM) from the fuses listed below is used for turn signal and hazard lamp operation.
• LT TRN/SIG fuse for the left turn signals
• CTSY/RF TRN fuse for the right front turn signal
• RR TRN/SIG fuse for the right rear turn signal
When the turn signal switch is place in either the LEFT or RIGHT position, ground from G300 is applied through the turn signal switch signal circuit to the BCM indicating the turn signal request. In response to this signal, the BCM applies a pulsating voltage to the front and rear turn signal lamps supply voltage circuits cycling the lamps ON and OFF. At the same time the BCM sends a GMLAN serial data message to the instrument panel cluster (IPC) to cycle the appropriate turn signal indicator ON and OFF.
Ground for the turn signal lamps is supplied as follows:
• G104 provides ground for the left and right front turn signal lamps
• G402 provides ground for the left rear stop/tail/turn signal lamps
• G403 provides ground for the right rear stop/tail/turn signal lamps

Do you have a factory scan tool (tech2) ? Your best bet , take it to the dealer .

Hi there., when flashers/indicators work rapidly on the dashboard you should check whether it is the same on the actual bulds. Fadt flashing is usually a sign of a bulb failure somewhere. If everything is functioning ok externally I would consult a competent technician for a diagnosis.

why skip RR lamp? what does IT DO? be like 1st 1st look.
dim, out, fast?

the rapid is talking to you
it means current is wrong,
shorted, or open lines. to the right.
its the wiring
if the bulbs are like 1157 brass base bulbs
why not replace them first , they can short inside,
asking online for guessing is silly, why listen to 10 examples for this.
but is cut wires.
shorted wires.
keep i mind the blinker asks funny for both opens and shorts
its a feature, that.
low current or too much.

the acid test is connect 2 TEST bulbs in parallel to the multi-function switch output right side,
blinks ok now, bingo bad lamps or bad wiring, from there to the right side.

if the hazard fails you never said????
if rigth front is dim or dead the bulb is blown or shorted
if new bulb there , then the wires are shorted.
to it.
the only help on line worth squat is how to test and use your meter.
you can, check for voltage
you can check continuity on that line (its just 1 wire)
and check for shorts on that wire to ground (lamps pulled right side to do that, lamps read 0 ohms cold)
look for collision damage RF? behind lamp sockets?
be first.
in my book.
looking is always first. sure.


I have your SM book open now, chapter 8w-50
the haz/blink module has just 2 pins, left and right out.
pin 3 on the module runs front and rear blinks, (a fact to know)
RF is BR/RD, brown -red stripe?
all comments USA on USA forum. USA car.
ok?

the right side lamp (sidemaker) only blinks turns if parks ARE oFF

Id try 2 new blinker lamps front and 1 rear.
that be first. or test the lamp on a battery , and no not get burned fingers hot wring a shorted lamp.

wish i knew
what rear did
and what both did on HAZ
out, dim or to fast.

Sounds like a burnt out bulb, that fast blink is to tell you a bulb needs replaced.

bad lamp or it is loose lamp replace lamp check connections