Last year we visited at the end of August into early September. It is a new thing for us having to deal with school vacation times. I was concerned, like you, that the weather would be too hot and humid and very rainy. I can't deal with humid heat.
I was surpised. Yes it was hot, and humid, but not as bad as I thought it might be. We wore light clothing, kept hydrated. It did rain, but not constantly - usually in the afternoons, the majority of the time it was fine for us. If the rain was too heavy, we would call it a day and go back to the vacation home and watch a movie or play games. I think we only did that once it two weeks. (It did rain more than that, but only once did we leave the parks early because of it)
Our dates were the 18th August and we were there for 2 weeks. That time of year is lighter for crowds because some schools are preparing to go back - so that is a plus.
All the parks have areas where you can cool off. Make use of stores, restaurants and of course indoor rides/shows for opportunities to cool off. We also had some personal misters and fans that we took with us just in case. Kids got to play in the water areas to refresh themselves.
It is not my favorite time of year to visit, but I would travel there that time of year again.
Hope that helps.
-Kay