I grew up in Florida without A/C.
Yes, we just dealt with the heat as best as we could. That meant no doing much during the heat of the day. It also meant dealing with cleaning the mildew off things on a somewhat daily basis. Example, we had our meetings on Sunday, Tuesday and Thursday. These were the days I needed to wear my "good shoes" ... not the ones I wore to school. Before each meeting I had to clean a layer of mildew off my shoes before putting them on.
You also got used to your clothes to feel "damp" when you put them on unless you hung something in the closet to keep them dry.
It is more than just the heat that makes people want A/C - having A/C sure does make life easier during the oppressive humid heat of daily life for 4-6 months running.