Does a humidifier cool the air?
You are quite correct about the thermodynamics. The heat involved in a phase change of water is so significant that professional HVAC guys specify the performance requirements for cooling as two different parameters: sensible and latent heat to be removed. The latter is the heat involved in phase change of water and is often more demanding than removing sensible heat.

"For example, in 1 kg of waste air at 20 °C with a relative humidity (RH) of 60%, the total energy is 42.5 kJ, more than half of which (22.5 kJ) is latent heat."

You can analyze your situation with a psychrometric chart like this:
http://www.truetex.com/psychrometric_chart.gif

Just pick your desired temperature and relative humidity and you can find out how much water exists as vapor under those conditions.