On our 2001 we have 2 fans, one manual mounted to the front of the engine in which case the thermostat that controlls it is a spring inside of the housing on the front of the fan and wouldn't have anything to do with your temperature sensor. The other one is electric and is controlled by the temperature sensor.
Either one could fail, the manual ones in particular become less accurate with age as does the sensors that control the electric one. Also on ours at least there are TWO temperature sensors, one controlls the fan and one works the dash gauge and reports to the computer. One can fail without the other failing so don't let that one fool you, be sure to find out if your model has one or two and check them both if it has two.
As for a headgasket leak, when your engine is toatally cold you can remove the radiator cap and if you can see it clearly and it's full, which it should be with a resivour/recovery system (if it's not that itself is a sign of a problem, closed systems should NOT have any air pockets) then you can start it with the cap off and watch it, if your headgasket is leaking you should see bubbles, watch it as it starts to warm, you'll even see the "flow" begin to get stronger as the thermostat opens, if you watch it to this point and then a few more seconds (the fluid will start to overflow a little as it gets warm, that's normal and not a problem).
If you watch it that long and do not see any bubbles then one final test once it's warmed to this point is have someone give it some gas and ease the RPM's up to at least 1700, the fluid will overflow more quickly but it should be a "smooth" overflowing, if you have continual bubbles than you probably have a head gasket leak.
One a Side Note: There was a Ford notice sent to dealerships in parts of the south and southwest suggesting that before selling V10 models in certain regions of the country that they should replace the factory installed
high-temp 192-195 thermostat with a medium-temp 180-185 degree thermostat to reduce the chance of overheating. Chrysler had a simlar notice and even went as far as sending out cars to the south with an instruction sheet and a new thermostat in the glove box of vehicles being sent to certain regions. The problem is many delerships never installed them and just tossed them in a corner during the new vehicle prep process. I'm sure Ford delaerships often did the same thing and I am wondering what about chassis or engines sent to outside companies such as Motor Coach builders? Did they ever even get such notices? Did they bother to do anything with them if they did or were they more worried about just getting the engines installed or the coach bodies put onto the frames?
If you change your thermostat you may consider going with the medium temperature one, it's still hot enough so the computer / ECM works fine and unless your going to Alaska will still keep you plenty warm during the winters. It will help keep your temps down on hot days. They do also make 160-165 degree low-temp thermostats but unless your driving in the Carabian those can sometimes cause the computer or ECM to think your always in "warm-up" mode so should not be used on newer computer conntrolled vehicles, though they can be great on older models without computers or ECM's such as carburatated models, especially in the deep south and southwest.