I meant to ask if you can hear the compressor clicking on and off in other settings. If it's clicking on and off, then it could narrow the problem down to a blower motor, resistor , or dash A/C switch.
But, now that I see that the heat works normally, maybe it's something else.
Does the fan blow normally if you have vent, defrost, floor selected?
I'm not an expert in A/C stuff, but maybe this info will help someone else figure it out. Sorry for the confusion.