I'm sure a lot of Floridians are chuckling over this thread. To us, June is relatively mild, both in terms of temp/humidity and thunderstorms. August and September are the absolute worst, but you can get mid-80's just about any time of the year in Florida. Our current pattern here in Miami is 60's at night, low 80's for high temps.
For packing, I personally would take a jacket to wear when the temps plunge into the 70's...but I'm a native Floridian.
More seriously, my number one suggestion is the one mtnman44 gave above -- hydrate, hydrate, hydrate. That is more of a health suggestion than a comfort suggestion. If you wait until you're thirsty, you've waited too long and probably will not catch up.
Once you get a little bit of heat illness, you're going to feel washed out for the rest of the day, and that will really lessen your enjoyment of your vacation. If you feel a wave of warmth sweep over your body, followed by a little weakness, I'd suggest taking the rest of the day off. As an EMT, I treat a LOT of heat illness patients, and that wave of warmth is reported by almost every heat exhaustion patient I treat.
If you have ANY kind of underlying medical condition -- respiratory or heart issues, diabetes, any deficit from previous illnesses or injuries -- the hydration advice goes double. Dehydration causes all sorts of chemical imbalances which could have serious effects for anyone with a chronic condition.
Even though we're used to the heat and humidity, our family follows the early/late theme park routine suggested above, usually with a trip back to the resort for a pool/nap break in the middle of the day. The downside of early/late in the summer is that the thunderstorms usually occur in late afternoon/early evening.
We also either eat a very light lunch, or nothing at all. Light snacking and lots of water and Gatorade seem to work better for us. A frozen lemonade may taste great, but it's the
volume of fluids that is critical. Avoid alcoholic beverages during the day, because alcohol is a big-time dehydrator.