As a Floridian I book 1st or 2nd week of June every year, done it for the last 10! It's the best time to go in Summer!
It's the coolest month of the summer, the humidity has not reached it's typical highs, so it's very tolerable, especially morning and nights. July, August & September, forget it, you sweat standing still!
The crowds are low since most of the northern schools aren't out yet. Florida public schools let out between June 1 & June 7 and the tour groups generally don't arrive for July and August, so the lines are manageable.
The big summer rains have not begun yet, and though June 1st begins hurricane season, they usually come later in the season.
We will be there in 47 days, yeah!!!!!!!!!!!!
We do go in August just before the start of school, this year Aug 20th, but it's way too hot for the regular parks, we do just water parks with all of the other locals!
Have fun!!!!!!!!!