Low beam lights don't turn off when ignition is turned off, what is the problem?

https://www.fixya.com/cars/t8706315-2003_gmc_yukon_xl_denali_low_beam

The headlamps may be turned ON in 2 different ways. First, when the driver places the headlamp switch in the HEADLAMP position, for normal operation. Second, with the headlamp switch placed in the AUTO position, for automatic lamp control (ALC). During ALC the headlamps will be in DRL operation in daylight conditions, or headlamp operation in low light conditions.
Headlamp ON/OFF control is determined by the body control module (BCM) by a signal on the headlamp switch on input circuit when the headlamp switch is in the HEAD position. When the headlamp switch is in the AUTO position, the BCM determines headlamps ON/OFF by the voltage from the ambient light sensor.
If the headlight switch is left in the ON position, the inadvertent power control feature will turn OFF the headlights approximately 10 minutes after the ignition switch is turned to the OFF position. If the driver places the headlight switch in the ON 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 turned OFF or the battery runs dead.

Low Beams - Standard
The low beam headlamps receive voltage from the headlamp low beam relay located in the underhood fuse block. The BCM controls the relay on the headlamp low beam relay control circuit. Low beam control is determined by a signal on the headlamp dimmer switch signal circuit from the multifunction switch. When the headlamp dimmer switch signal circuit is open, the BCM grounds the headlamp low beam relay control circuit, illuminating the low beam headlamps. The right and left low beam circuits are independent are fused independently in the underhood fuse block.


Do you know how to test electrical circuit's - relay's ? How to use a digital voltmeter ?

Do you know anything about automotive electronics ? Electronic control modules .
Headlamp ON/OFF control is determined by the body control module (BCM) by a signal on the headlamp switch on input circuit when the headlamp switch is in the HEAD position. When the headlamp switch is in the AUTO position, the BCM determines headlamps ON/OFF by the voltage from the ambient light sensor.
If the headlight switch is left in the ON position, the inadvertent power control feature will turn OFF the headlights approximately 10 minutes after the ignition switch is turned to the OFF position. If the driver places the headlight switch in the ON 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 turned OFF or the battery runs dead.
The high beam indicator is illuminated when the instrument cluster receives a GMLAN serial data message from the BCM that the high beams are illuminated.
High Beams
The high beam headlamps receive voltage from the headlamp high relay located in the underhood fuse block. The BCM controls the relay on the headlamp high beam relay control circuit. High beam control is determined by a signal on the headlamp dimmer switch signal circuit from the multifunction switch. When the headlamp dimmer switch signal circuit is grounded, the BCM grounds the headlamp high beam relay control circuit, illuminating the high beam headlamps. The right and left high beam circuits are independent and are fused independently in the underhood fuse block.
Low Beams - Standard
The low beam headlamps receive voltage from the headlamp low beam relay located in the underhood fuse block. The BCM controls the relay on the headlamp low beam relay control circuit. Low beam control is determined by a signal on the headlamp dimmer switch signal circuit from the multifunction switch. When the headlamp dimmer switch signal circuit is open, the BCM grounds the headlamp low beam relay control circuit, illuminating the low beam headlamps. The right and left low beam circuits are independent are fused independently in the underhood fuse block.

Hooking up a factory scan tool an checking DTC'S - diagnostic trouble codes would be the very first step in trouble shooting this problem .
DTC B257A 00: Headlamp Switch Input Signal Mismatch
DTC 2575 00: Headlamp Control Circuit

Those lights are on different circuits. The high and low beam lights are powered thru the headlight switch, the turn signals, brake, and hazards go thru the turn signal switch.
I would want to know if each circuit has power at the switch that controls it.

one the your light bulbs have a broken contact or the ground shorted out

ck for power to bulbs ck fuses ck switch

wire a relay parallel into the low beam light power wires
run a fuse controlled wire from the battery to the relay
when in low beam the relay activates to turn on the lights
when in high beam there is no power to activate the relay so the lights go out
If you read the regulation carefully , you will see that is also required to have a separate on/off switch for the lights

Sounds like the low/high beam relay switch is not functioning properly. The relay should turn both high and low beams off when the ignition is turned off.

Common problem, there is a headlight relay in the power distribution center under the hood that needs changed out.

i think im right in saying the low beam bulbs are twin filament(low and high beam in one)and the main/high beam are single filament(spotlights).