We go every year during the last week of August, first week of September. "Crowded" is a relative term. I would say the crowds are manageable, but they are there. Sure, the Southern schools are in session, but there are plenty of folks visiting from the other states!!
We make use of EMH so for us the rides will have little or no wait at that time. Later in the day, FP return times will be an hour or more. Counter service is usually not crowded and we always have PS for sit-down meals. The resort pools are, IMHO, crowded in the afternoon because everyone seems to have the same idea. The busses run very often and we frequently get a seat.
For us, these two weeks are the best and only way to go because pulling the kids out of school is not an option in our school district.
I cannot recall being rained-out at any time. The showers seem to fall, briefly and heavily, while we are inside an attraction or restaurant.