2004 trailblazer ext high beam headlights wont turn off and the relay makes high pitched noise

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.

Low beams don't work ??????? But high beams do ????? DRL - daytime running lamps ?

HDM relay # 46 low beam relay , under hood fuse box.

You can't turn them on manually with the switch . You have the switch set on auto ?
Headlamps
You can use the headlamps in two different ways.
• Place the headlamp switch in the ON position for normal operation.
• Place the headlamp switch in the OFF position for automatic lamp control (ALC). During ALC the high beam headlamps are On at reduced intensity for daytime running lamp (DRL) operation in daylight conditions, or low beam headlamps On in low light conditions.
The DRL/EXT LTS fuse and the EXT lamps fuse in the underhood junction block supply battery positive voltage at all times to both of low beam lamps and to the right high beam lamp. The low beam circuit continues to the high/low beam switch, then, depending on the position of the dimmer switch, connects to the normally-open headlamp relay switch contacts. Placing the headlamp switch in the HEAD position provides a ground path for the headlamp switch input of the Body Control Module (BCM). After receiving this input, the BCM energizes the headlamp relay. When the headlamp relay is energized, the low beam circuit is connected to ground G201. The high beam circuit is divided into a left and a right side circuit.
The DRL/EXT LTS fuse in the underhood ignition block supplies battery positive voltage to both the coil and to the normally-closed contact of the DRL relay switch circuit. During high beam operation, the left side current flow is through the closed contacts of the DRL relay, and to the left high beam lamp.
The EXT Lamps fuse in the top of the engine fuse block supplies battery positive voltage to the right high beam lamp. Both high beam circuits are then connected together, and continue to the dimmer switch. When the driver closes the high beam contacts of the dimmer switch, the circuit is grounded at G201.
Both the headlight switch and the BCM provide a ground, or control circuit, to the headlight relay coil circuit for manual or automatic headlamp operation. When you place the headlamp switch in the OFF position, and the ambient light sensor indicates low light conditions, the BCM energizes the headlamp relay for low beam operation. You may switch the headlights to high beam by pulling the dimmer switch handle toward you until the dimmer switch is activated. When the dimmer switch is placed in the HIGH position, the indicator lamp input to the BCM is pulled low. 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 headlights 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 the ACCY position, the headlights will remain ON until you turn them OFF or until the battery runs dead.

The body computer controls the headlight relays based on input from the switches. There is a relay for high beam and one for low beam in a power distribution module in the engine compartment.
I don't know if the relays are available without buying the entire module. One of them could be stuck on.
You may need a shop manual to get the wiring diagrams and procedures to test the circuits.
In the mean time you should be able to pull the 50amp fuse for the headlights.

try your high beams. second one is probably the high beam.

The daytime running lights run the high beam bulbs at low voltage so it sounds like you need to put the relays back where they were and either replace the low beam bulbs or repair the wiring to them.Also the relay you removed could be defective.

headlights have burned out on the low beam side, replace both headlights, should fix your problem

First face the vehicle so the battery in on the right.
Second open hood.
Third Remove the Grill by pushing firmly on two retainer clips that are on the top, use the wide slot screwdriver to get the rest of the clips off. Pull up gently on the grill it should then pop out. The grill extends under each headlight assembly any other retainer clips should release when the grill in popped out.

Step 2 Fourth on top of the headlight assembly you will see one 1" tab on the right and left side, pull the tabs upward about two inches and this should release the headlight assembly. You then should be able to pull the headlight assembly out toward you.

Step 3 Finally at the back of the headlight assembly you will see two rubber caps one for low beam and one for high beam, the white base is for the low beam and the black base is for the high beam. Unsnap the wire harness from the bulb base, turn the bulb clockwise.

Replace the bulbs high beam on the outside and low beam on the inside.

the click you are talking about is the high beam relay.
you nedd to get a tester or voltmeter and check the sockets for 12v.
there has been instances where both low beam bulbs went at once.
do the daytime running lights work?
if not,supect the daytime module...it is also wired into the parking brake.
if you put the park brake on first,then start the vehicle,the dtr do not come on

Try swapping #45 cooling fan relay and #46 HDM (headlight driver module) relay in the underhood fuse block. These are solid state modules and are not like conventional relays.