I have never been to DL (going in August for the first time), so I can't speak to the heat in CA, but I've been to WDW many times. Our last trip was June of 2017, and it was plenty hot! We used Mission cooling towels, and I highly recommend them for beating the heat. Every morning before we left our hotel room, I would soak one for each family member with cool water (I got 4 different colors, so each family member used the same one every day of our trip) and I put each cooling towel in a separate ziploc bag. They didn't take up much space in the family backpack; we pulled them out as needed, and snapped them (put an end of the towel in each hand and pulled the towel in opposite directions) 2 or 3 times, and the towel cooled considerably. You can wrap them around your neck, drape them over your head, or whatever feels best. The best part is you can bring them into a bathroom and re-wet them with cold water and snap them over and over again throughout the day. Each night, I would take them out of the ziploc bags and hang them up; they dry quickly that way, and then they would be ready for use the next day. If you're doing a load of laundry while on vacation, you can throw them in and they wash up really well too.
I've gotten them at Target in the athletic/sporting goods section, and at Dunham's Sporting Goods. I'm sure you can find the Mission towels lots of places, but those were the stores close to home for me.
Good luck staying cool!!