It maybe the sender unit it is located in bottom hose outlet on the radiator. Cheap to replace. Check wiring plug is connected to it. As you stand looking at engine bay it is at bottom right hand side of the radiator. Mine has no engine cover underneath so i just jack it up front passenger side, axle stand, to give room to get head under front bumper. If you look up at rad bottom hose you will see the sender mounted in the plastic.
I had prob with my fans running all the time even with ignition off i had to replace the fans as the fault was with the main fans internals. Got pair from breakers yard been fine ever since. However my temp guage was working.
It maybe as simple as a broken wire. Ten minutes is also a long run on time and could put strain on your battery
Hope some of that helps