You can only have a net outflow of water if the air is not saturated with water, which is a function of the air temperature. If it is saturated, a dynamic equilibrium will establish, where water molecules from the air jump back into the surface of the liquid film.

That's why a 100F day in Arizona (dry!) will be much more pleasant than a 100F day in Florida (humid!) - if it's dry, your body can still get rid of the heat. If it's humid, it will just 'stick to your body'.

At equilibrium, water condenses as fast as it evaporates. So while the fan is blowing water molecules away from you, it’s also blowing water molecules toward you, which condense on your skin. Net heat transport would be zero.

The wet bulb temperature already takes this into account. If you wet your body and sit in front of a fan, your surface skin temperature will be very similar to the wet bulb temperature. Once you know the skin temperature, everything exterior to your body has already been taken into account and it's a question of how many watts of thermal energy can be transferred from your core to your skin.

When your body is warm (i.e. not trying to conserve heat by reducing circulation) then this is essentially linearly related to the difference between core temperature and skin temperature. When the wet bulb temperature is above 95F, then you create more thermal energy than can leave your body when your core is at 99F, so your core body temperature rises and all the bad things that go along with it progress.

Yes, you are, sort of. The wind will still cool you, but only down to the wet bulb temperature (that's sort of the definition of the wet bulb temperature). So you will no longer be able to keep your core temperature constant, it will rise to lethal levels.

Thanks for the explanation. I'm still puzzled.