i can tell you with 95% certainty after having owned one of these vehicules, the stuck door is a wild goose chase, its low on refrigerant.
How can it be low if one side of the car is cold? So the stuck door is out! I still think it's the tube that carries the cold air to the vent came loose or fell off!
The tbe happens to carry vacuum tho..... Very common occurance on the grand pricks. It's usually due to newbie mechanics thinking that all door/duct functions are electric motor controlled. Nope, GM still has it's head in the sand and still uses vacuum.
