You're wrong, your spouse is right.
Thermal energy is required both to raise the temperature of a mass of (in this case) water, and additional thermal energy is required to change its state from liquid to gas. This additional thermal energy is spent without creating any actual temperature increase, but it had to come from somewhere.
In this case, the thermal energy for the state change came from the surrounding air. The energy didn't come from changing the state of the air, so it must have come from lowering the temperature of the air.
As others have noted, this only works in low-humidity environments. If the air is already saturated with water vapor, no more evaporation will occur. This is why high-humidity environments feel hotter: your sweat isn't evaporating to cool you off.
This meme is stealing.