Im not on this site often so I may not be back for a response soon but:
It sounds like you are having a headlight gear problem. The gear inside the headlight motor is made of plastic and frequently the larger one strips. Telltale signs are possibly a chattering noise when they go up or down, or the sound of the motors continuing to run after the light is up or down (normally the increased resistance in the motor when the light unit hits the stops when opening or closing is what causes the circut to shut off, if the gear is stripped the motor sees no increased resistance so it runs on) or what you are seeing (one moves and the other one doesnt).
I noticed that you said "
When I start my car in the garage the headlamps turn on as expected and close after a few moments on the road." if that means you have the twilight sentinel feature enabled, and the lights are popping up and down automaticaly as you drive in and out of dark areas during the day, well you gotta know thats not going to make the problem better

. Many people disable that feature (can be done right from the DIC on the dash, its in Options, just turn it off, saves wear and tear on the motors.
If you dont have twilight enabled and they turn off and go down after driving a bit, it could be that the bouncing from the road is moving them a bit and then they grab.
The light motors arent bad to take out, its basicaly four screws, two bolts and a pin and you have the light units out (lotta room in there), and the motors have two bolts holding them in.
If you find the gears are bad (and you're the only one who is going to be able to tell, based on what you see and hear when you turn them on/off).
You can either get replacement gears (usually brass ones) from some vendors (sorry dont have the link atm but they shouldnt be too hard to find) but that involves cracking the motor case and resealing it with epoxy, which mostly does but sometimes doesnt hold.
Or you may have to replace the motors. You may be able to get salvaged ones from parts places or bite the bullet and go new.
Good luck
