I’ve always likes late August because crowds are lower. It does tend to be very hot and humid, but we do the early mornings, then a midday break/pool time, then head back for the slower evening hours. With older kids you can hang for the later nights. I try to limit the times we follow a late night with an early morning.
Heavy rain is a possibility, but if there’s particularly tropical weather predicted I’ve found that just makes the crowds a little lower. One trip we went early because there was a tropical storm, possibly hurricane predicted, but it fizzled over the Caribbean and southern tip but sucked all the moisture out of the air so we had one day of rain and that was it. Another trip a hurricane actually made landfall just north of Tampa, and we had one night of bad winds and a full day of downpour and that’s when Crocs and dri-fit clothes really come in handy, because the parks were largely empty that day. As long as you can be flexible and your trip isn’t just a 3 day in and out affair then even a missed day for weather is mostly balanced out by the lower crowds.