Another Minnesotan here... chiming in.
For the past 40+ trips I've been on, they have all been at the end of August. (well, except for that one year that the pilots were on strike and then we went Mid-Sept) And honestly, yeah it's hot. Now we get heat and humidity here in MN too, so that makes us sort of used to it. But our plan was if we got to hot, we went inside where there was AC, got something cold to drink (or ice cream to eat) and took a break from it. Heck, the last time we were down, our DD was 6. She wanted to stay at EPCOT all day. And we did. It was probably about 98 degrees all day with 70+% humidity. We made it around the park and then we were going to go back to the hotel and relax for a while before heading back. She had a fit that we were leaving. So we went out and got on the monorail. Just rode it around and around. Well, she fell asleep! So we stayed on for about an hour and then woke her up. It totally recharged her batteries. She was then good until we left the park after Illuminations.
As far as shoes, I wear a lightweight pair of New Balance tennis shoes/sneakers. I have found the ones that are sort of the "mesh" style work really well. They allow your feet to be cool, but give you the support for walking.
And yes, lots of water.... lots of sunscreen. Even if it's cloudy. I made the mistake one year of saying" Oh we got the kids lathered up, but I'll get to it in a few minutes." BIG MISTAKE! I was Sebastian red for about a week. (man did it hurt). Crappy thing was, I never tanned. Still Casper white.
The afternoon showers were always my favorite. And yes, you can almost set a watch by them. Every day between 3-5pm. So that is the time we usually had some sort of meal... just to get inside and let the storm subside. Right after a storm is a great time to be out. Still cool enough from the temps dropping, and no humidity, or very little. If we got stuck outside during a storm, we had some cheapo ponchos from the dollar store in our packs. I think during the 16 days were were there last time, there was only 1 or 2 rainshowers where we actually had to put one on.
So we're going the end of August again this time, and we love that time of year. Yeah, the heat and humidity can stink big time (and so can it stink from everyone sweating) but hey, I'm on vacation, at my favorite place in the world.... I'm not going to let it stop me!!!!
~D