Hello Chris,
as you have correctly suggested fans only move the air around, they do not cool it, however still air acts rather like an insulator so you can feel hot, but introduce some movement and you sense that you are cooler. The new air that comes in to contact with your skin does take more heat away than still air.
the "coolers" that you can buy where they evaporate water in the air flow, actually humidify the air, and so they will reduce the air temperature, BUT the increased humidity may become uncomfortable.
Only true air conditioners, where the heat is "exchanged" will make the air more comfortable, by reducing the temperature and reducing the humidity.