Going away from sweat for a minute:
Put a pot of water on the cooker and boil it. Once boiling the water is 100C, but it doesn’t all turn to steam immediately. That is because it takes a LOT of energy to turn liquid water into water gas. Significantly more than the energy required to heat it to start with. It takes more energy to go from 100C water to 100C steam than it does to go from a block of ice just above absolute zero all the way to water at 100C.
Water doesn’t need to be at 100C to transition to being in gaseous form, it just needs to not be in an equilibrium state with moisture already in the air. As long as the surface of your skin is warmer than the dew point of the water vapor at that temperature in the air some water will evaporate from you, taking a lot of heat with it as it transitions from liquid to gas, called the latent heat of vaporization.
The less moisture in the air, or the colder the air to start with relative to your skin, the more easily this process happens, which is why a “dry” heat is more bearable than an equivalent humid heat, and also why a dry-cold feels a lot colder than the equivalent temperature when the humidity is high.
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