There is something you can do - but you will absolutely hate it at first: acclimatize. This is going to be kind of a long post while I explain...
I grew up in Florida, and lived all over the South - in the age of great A/C. So while I have always been used to the heat - I can really only take it in short bursts. But when we moved to Okinawa, our first summer there was one of the hottest they had had. Our house didn't have central A/C - just a couple of individual units (bedroom, tatami room), which ran constantly, but never really cooled things down. (82-85 degrees F) That first summertime electric bill was over $800 dollars American. We were beyond horrified at the prospect of 3-4 more months of this.
A friend of ours suggested acclimatizing. He and his wife had done it. In short, they eliminated as much air conditioning as they could from their lives. No AC in the car, no AC at home. He told us it took them about three weeks. Because of the very high humidity, they ran a dehumidifier every few days to get the moisture out of the house.
So we tried it. The only AC we used was at bedtime - but at a warmer temperature than we used to set it at. For daytime and riding in the car, we threw all the windows open. Work, shopping, etc. was unavoidable, but we lived without AC every chance we could.
The first week was wretched. I think I changed tops 3-4 times a day, and bottoms at least once. But it got easier over that three week period - and, just as our friend told us... we acclimatized. We were at a cookout about a month later, and it was in the 90's. Everyone around us looked ready to melt - and we were surprised to find we were barely sweating. So it really does work.
Our electric bill came waaaaaay down, and we didn't have any problems with molds because we ran a dehumidifier (it ran constantly the first 48 hours, then as it got the moisture out, we only had to run it every 2-3 days).
(Over the course of several years, the units were replaced with new, energy-efficient ones, and we would run the AC during the hottest part of the day, and at bedtime.)
I hope this helps!