First, Let me say that the radiator fan will have no appreciable effect on temperature if the engine is not running. It pulls air through the radiator when the car is running but not moving to remove heat from the circulating coolant (water/antifreeze mixture). Your symptoms are really screwy. I would check to see if your system has enough coolant and add as needed. If that is not the problem, I’d have the entire system checked for blockages and flushed, and replace the thermostat for good measure (they’re cheap).
If things are working right, you might see a slight rise in the temp. gauge after a fully warmed-up engine is shut down as the coolant (which is now not circulating) will absorb some heat from the metal, but it is really a net temperature loss. Once the engine is shut down, it stops generating heat.