If the motor current checks out ok then there may be an air flow problem causing the trouble. If the blower motor has never been replaced it may be a good idea to replace it anyways.
The speed control module needs air flow to cool it so check for things like a mouse nest in the ducting. The speed controller I am refering to is a solid state device that has perhaps 3 or 4 lead connections and fits inside the air duct. I am not talking about the main controller that fits in the dash with the controls on it. Here is a link to what I am refering to.
http://www.partsgeek.com/catalog/1992/buick/lesabre/climate_control/blower_motor_resistor.html