Drink constantly; if you wait until you are thirsty, you have waited too long. We use Brita bottles and fill them in the parks from the water fountains; we have never gotten sick from the fountain water. First stop in the MK is by the lockers under the station, some really well-cooled fountains there.
Wear loose light-colored clothing, and slather on the sunscreen, protected skin is cooler skin. If your skin starts feeling hot, you are probably burning a bit. If at all possible, get Circulair fans; they have a wide opening, so you can put a few ice cubes in the water; makes a huge difference in how effective it is at cooling you down.
I've done the hand towel thing, too, but I now know how to do it one better -- a flat cloth diaper! (Use the kind that is just a rectangular piece of cloth, not the kind that is folded over and stitched.) They are soft, very absorbent, and much less bulky, and so much easier to put around your neck or store away when not in use. They dry very quickly as well. Most people have no idea what a flat diaper looks like these days, so you won't get any funny looks, it just appears to be a white rag. These also work well for shading babies in strollers; the cloth is thin enough to let plenty of air through.
WEAR A HAT with a brim. It is important to keep your head cool to lower the risk of heat exhaustion and/or sunstroke. (I know that sounds like a joke, but as a veteran of a couple of sunstrokes, I assure you, it isn't!) Going red in the face is not a good sign, but if you see anyone in your party suddenly go -white- in the face, stop whatever you are doing and get the person into a cool area immediately, and don't listen if they say they are fine. Apply cool (not cold) water to the skin, especially at the back of the neck, and watch for tremors and/or disorientation; if you see them, tell a CM you need paramedics. Not sweating in high heat is a BAD sign! It is a complete myth that only children and old folks get heat-related illnesses; people in perfect health can get it at any age if they overdo, and the damage can be lasting.