I live in Louisiana, and this is something that has come up a LOT. We had a mild summer this year, with it only breaking over 100 a few days, and the humidity has largely not been "so thick you can see water" as well. I've seen it 120+ degrees with humidity in the 90% range.
Basically, you just suffer.
What we do is we go back to wearing clothing from pre-eletricity coming to the area. Kilts and breechclouts don't care if it is 110 degrees or 10 degrees. You wear looser clothing, and you watch a lot of people wearing a lot less clothing than is proper up north, to to ward off over heating. It is actually illegal for the power company to shut off your power in the summer and the winter here, because of how many people die from exposure.
We also do not get a lot of wind. I you feel wind, a story is blowing in, and the heat and humidity can spike for up to several days before hand.
Traditionally, down here, homes are made with high, high seelings, with large windows that stretch almost floor to ceiling whenever possible. You also see dog run house, which separates the living quarters and the cooking portion of the house by essentially a covered porch. The Romans/Italians use unglazed tile, that they pour buckets of water on, and use the breeze to make a swamp cooler. You hear of similar being done with dog runs as well.
Also, if you have a cotton bandanna, if you get it wet, you can snap it through the air, and it will chill for at least a short amount of time.