I do not have practical experience with this, but I think that I remember that what you are describing can be done. This is a variation of an adiabatic saturation process. If I remember right, an adiabatic saturation process moves along a line of constant wet bulb temperature along the psychrometric chart towards saturation. When the water is chilled, I think that it changes the slope of the process on the psychrometric chart (to the left of the wet bulb line), resulting in lower humidity and lower dry bulb. I can't be certain, but I think that this is right.

What you are saying about humdification also sound right; I think that the slope shifts in the other direction towards higher humidity and dry bulb (to the right of the wet bulb line).

Jeff1
Depends if the temperature of the coil that is cooling the air is bellow the dew point of the air-vapor mixture. If the temperature of the cooling media is less than the dew point, the vapor in the air that is in contact with the cold surface condenses. When the media es colder the drying efect is larger. The drying effect depends on the depth of the cooling coil, the numbers off fins per inch ( both influence the by-pass factor of the evaporator) and also on the TD (Ambient air temperatura-cooling media temperature)
On the other hand when you heat air you make it more higroscopic, that is the reason because you use hot air for drying.