The engine management puts at least one of the radiator fans on as soon as you put the air con on, and switches off when the condenser has cooled down, even if the engine is off, so you CAN hear a fan slowing down when you switch off.
The compressor is engine driven, it stops as soon as the engine stops.

The control of A/c is done by engine management, either the A/c compressor clutch is stuck or the control of it is U-S.