"Fanning" as in you can see it spinning - or - "fanning" as you can feel the hot air moving?
Clutch fans usually operate at a percentage of the engine speed - up to a point - when they reach that point (a few thousand rpm), they don't go any faster, regardless of how fast the engine turns, or how hot it gets, and when they fail, that rpm drops.
A failed fan clutch will give you some on the symptoms you're getting - once the vehicle is moving, and the engine load is light, the forward movement forces air through the radiator, and will usually keep the engine cool, reduce that forward speed or increase the engine load, and you can tip the scale the other way, and now you need the fan to move more air to keep the engine cool.
Unfortunately the symptoms are not conclusive, a partially clogged radiator will have a similar effect.
Before starting the car tomorrow morning (ie with the engine cold), open the hood and try to turn the fan, it should turn but with a fair bit of resistance, and should stop as soon as you stop - if it spins freely, it's dead or dying - repeat the tes with the engine at operating temperature (but not overheating), it should turn with less resistance, and this time might spin a turn or two when you stop turning it. I know this might sound backwards, but with the engine cold & off the fan actually gets harder to turn, once the engine start and the fan turns for a couple of minutes it will free up considerably, and then become harder to turn as the temperature rises - ever notice that you can hear the fan quite loudly for the first 3~4 minutes after a cold start?