[Bob Kontak]

Not an expert but let's say when you switch the AC on (and the fan stops blowing) the compressor in the engine compartment kicks on properly and does it's thing and sends compressed refrigerant into the coils that get cold in the front where the blower is. The blower does not work so the coils get real cold and maybe even freeze up some. You turn the AC off and your coils are real cold and the fan kicks on and you get cold air for a while.

I would see if the compressor in the engine compartment kicks in when the AC button is pushed to on. The belt will spin if AC is turned off but the center circular part will not. When turned on it should spin.

If that is working then there has to be some issue with the switch and the wiring associated with the blower fan.