Car runs hot?

Hi, I have a 2002 S Type Jag. 20 mins. After driving the temperature gauge starts to fluctuate. The car runs however I can see the temperature gauge moving up as I slow in traffic or idle, once I take off on the open road the gauge moves slowly ( the fluctuation happens through the drive time) back to the normal position in the middle. I’ve had the thermostat checked and told that was not the problem. Also, I notice that the fan does NOT spinning when the engine has been running for an extended period of time, I hear an exhaust sound as if air is being sucked away for the radiator. Can u help?