Drying out is what they're supposed to do. The evaporative effect is what keeps you cool. The water absorbed in the clothes actually keeps the 'heat' in. If the air temperature is below body temperature, the moisture evaporating off your skin carries excess body heat with it.
In a hot dry environment, where the ambient air temperature is approaching or above body temperature, a wet shirt can help because that moisture is absorbing the heat from the air, before it gets to you.
​​​​​​In a very humid environment, the moisture has a harder time evaporating. A synthetic shirt will carry sweat off your skin, and also give it more surface area to eventually evaporate. A cotton shirt will absorb this moisture, and the excess body heat that comes with it, and not release it to the air. Have you ever noticed how heavy a wet t-shirt gets, or how long it takes to dry out?
I've spent most of the summer working in Louisiana, wearing FR-rated (100% cotton) work clothes, and it has been miserable; they stay wet for hours


There's a reason that there are almost no clothes intended for active/athletic use (like cycling) that are 100% cotton, and it's not marketing.