Okay so background here- my husband and I are from a northern state with all four seasons, moved to Florida at one point and found the summer unbearable, and happily reside in a state with a short "hot" season.
All that being said, we still choose to go to Disney on September. Rope drop is your friend, as are dri-fit type clothes. We go back to our room (we like to stay in the Epcot area) for/around lunch, and go back out for the evenings. During the lunch break, one crucial thing for us is to shower and change clothes. It is utterly refreshing as is a possible catnap. This does mean we bring luggage that is excessive for the length of our trip, or do laundry even on a short trip.
One other thing I have to personally handle is making sure I eat enough "real" food throughout the day. I love the little fruit and vegetable snack stands in the parks to get a little snack of carrots and celery instead of popcorn. Just something that helps me personally.
It usually rains in the afternoon in September (hurricane season) and this is also a good time to be either handling indoor stuff or resting in your room.
You can do it! Seriously, we despise sticky weather and we deal with it for the lower crowds and the better room pricing.