The water may be circulating at tickover and returning to the header tank, but you've already said the temperature goes up the faster you drive. This screams water pump to me. The faster the engine is going, the harder the water pump has to work and thats when the impellor slips on the shaft. Or the impellor is half missing so it can't move the coolant fast enough at high rpms.
If you want to know for sure without stripping the cam belt off to get to the water pump you can remove the thermostat and just about get your finger on the impellor to see if its in one piece and to try to turn it. Chances are some of it will break off in your hands they get VERY brittle with age.
temp sensor faults usually make the needle flick up and down randomly, quite often dropping down to 0 while you drive then straight back up again. also the old style temp sensors were black.. i think the green ones replace them, couldn't say 100% without checking.
