I am planning a trip, and it does get crazy trying to find the perfect time. Our requirements are; not too hot but still able to swim, not too crowded, and not insane pricewise. The best time I have found for all this is mid April. Fall would probably work too, but we'd rather do spring. I personally think April, after Easter week, would be a really good time for you. Good luck!
As far as weather, Fl weather is not super predictable at the best of times, but April is before the really rainy season starts, and yes it is technically hurricane season, but so is most of the year. So, this would be as good a time as any.
Final thought, if the weather is a major factor, maybe DL would be a good choice? Just a thought. It's too far for us to go right now, but I'd like to someday.