Here are my tips:
(1) Hydrate, hydrate, hydrate. You can get free iced water from any Disney counter-service location. Use it to refill your filtered water bottles often.
(2) If you don't feel like carrying a "mister" around, take advantage of the wonders of evaporation the way my mom used to do with our family, in the days before personal cooling devices were invented and before "splash zones" existed at the parks: take off your shirt in the restroom, run cold water over it, wring it out thoroughly and wear it until it's dry. Repeat as necessary. (*Note: When using this trick, be sure to select shirts that won't be semi-transparent when wet!)
(3) Build frequent breaks into your touring plan. When it gets uncomfortably warm, buy a frozen treat, or duck into an air-conditioned shop or attraction.
(4) Plan your routine to ensure that you're out of the sun during the hottest times of the day. Get up early, go to a park with morning EMH before rope drop, tour it ("like it's hot," LOL) and leave before lunchtime, when the sun is at its zenith. Have lunch at your resort in an air-conditioned venue and then take a long siesta and/or swim break until dinnertime, when the temperatures will start to cool once again. Now refreshed and rested, tour until late. When deciding what attractions to do when, schedule outdoor shows or waits in unshaded/outdoor standby queues for first thing in the morning or after dark.
(6) Dress for the weather: focus on the fabric, rather than the "coverage" of an item. For example, a linen or Coolmax shirt (even one with long sleeves) will be far more comfortable in the heat than a cotton-and-polyester tank top, which will trap heat and moisture. A light-colored hat can help deflect the sun, and obviously sporty sandals will be cooler than heavier footwear. (I'd avoid flip-flops, Crocs or other footwear that is slippery-when-wet, especially if you have sweaty feet or plan on visiting a splash zone to cool down.) And of course, wear sunscreen!
