One thing I can't stress enough... KEEP EVERYONE HYDRATED!!! I can't tell you how many times last summer I'd see someone collapse on the Monorail platform due to dehydration.
On the Monorail Platforms at the TTC & Epcot, we have water fountains (The one at the TTC/Epcot platform has the coldest and best free water on property!). All too many times I'd see a kid come up and start to drink, only to have the parent drag them away to wait for the train.
PLEASE folks... If your kid wants a drink of water... LET THEM! Sure, they might have to use the rest room every hour... but trust me... keeping them properly hydrated is more important.
I know bottled water at Disney is a bit pricey... but I've seen a lot of people bring regular water bottles with them... and then re-fill them at the water fountains. Also, if your room has a freezer, fill it with water, and set it in the freezer overnight. As the day goes by, the ice will melt, and you'll have nice cold ice water... (just beware, it will sweat)
As for keeping cool, and avoiding meltdowns... Try planning your day by alternating inside and outside attractions. Also, (and I'm not trying to plug merchandise) stop into one of the gift shops every now and again... a lot of them have really good A/C. In addition... some of the quick service eateries are well air conditioned... stop inside and sit for like 5-10 mins and discuss your plans for what attractions to see next. When watching parades, find a nice shady area if possible.
Maybe mid-day... take a break from the parks. Ride back to your Disney resort, and hop in the pool for a bit. And then head back to the parks when it cools down in the evening.
Hope that helps. And I'm sure others will have other great ideas