It is impossible for the engine to get hotter after you turn the engine off as illustrated by the first law of thermodynamics.
However if you have been driving, especially at speed, then you have gone from a situation of high air flow, high coolant and oil flow to suddenly zero of all of these. Therefore all the heat in the hottest part of the engine will transfer to the cooler parts and the coolant, so parts of the engine will get hotter after you stop! As will the engine bay.
I don't know where the temp sensor is for the radiator fan but they are often no longer on the radiator as they commonly were. If its on the engine then it could well trigger after engine shut off.
However on my diesel it is extremely unusual for the radiator fan to come on unless the aircon is on and you are stationary.
It would not hurt to check or just replace the coolant temperature sensor. They do fail and they do behave erratically towards the end of their useful life.
Some of them have separate outputs (more than two pins on the sensor) for the ECU and the instrument cluster and one or both outputs could be faulty.
Best thing to do is have a look around the engine bay and see if you can see a sensor on the radiator or on the cooling system somewhere else (or maybe it has both). Check the wiring and connectors and test with a multimeter.
As it has been said, problems with the MAF sensor can trigger the fan to run as you describe. Have you tried the advice on that?
Something like this
http://cpc.farnell.com/duratool/d030...BoC5i0QAvD_BwE
may be useful for verifying temperatures in the absence of being able to access a diagnostic computer.
If you coolant temperatures really are too high then you know you need to overhaul the cooling system, if not then you need to go testing sensors and chasing electrical faults.