You can hand wash them in a bucket with washboard (though we didn't) and line dry them. We actually line dried ours about once a month to air them out and sun-bleach the cloth. About 30 $5-15 cloth diapers lasts you 3-4 days between washes.
I've lived with people who did that. You put a used diaper in the toilet, and do a couple stir and flush cycles. Then you put them in a bucket with detergent. Once a day, you hand wash those, and hang to dry. With about 20-30 diapers total, you can manage.
But damn, finding used diapers in the toilet is annoying. Plus wet diapers hanging all over to dry. If it's your kid, though, it's not a big deal.
In my area, where relative humidity is often 60-90%, this form of drying would result in moldy cloth, not dry diapers. My guess is Tampa's microclimate is probably similar and line drying isn't an option.
How do you think people washed and dried clothes there 100 years ago? How do you think they do it in more humid and poorer places today? Do you leave your towel on the towel rack and let it dry there or do you throw it in the dryer every time?
Mold thrives in moist, dark places, not most sunny ones.
We used cloth diapers in a place that's hot and humid but seasonally cold (the US Deep South). Outside on the line, diapers dried a lot faster in the summer than the winter. We ended up having to buy a used dryer for winter use, because they simply didn't dry fast enough to be ready by the time we needed them.