June isn't as ugly hot as like, August, but it gets worse the deeper into the month you get and regardless it's hot and you'll want to have a gameplan for coping. Personally I never found
cooling towels to be all that effective, I have tried a few brands and all they really did was make my neck sweaty after a while. After trying a few different things on my trips in summer 2021 by far what helped the most were small neck/handheld fans and a
UV umbrella. I know many are not fond of carrying umbrellas in the parks but they make travel size ones that aren't huge, I got mine off
Amazon and it fits in my Loungeflys. I find that even with the humidity, taking the blaring sun out of the equation makes things way more bearable, because there are SO many attractions where shade is next to non-existent in the outdoor queues and if you get stuck in the wrong part of a queue it's Bad. Specifically thinking of Spaceship Earth, when I went in August 2019 and the FP+ queue was insanely backed up and I was stuck with the hot sun beating down and had to grab a Minnie Van back to POFQ when I got off the ride because I felt so sick. Didn't have any issues July 4th weekend 2021 even in Epcot with my umbrella and neck fan, by comparison. It felt a good 10 degrees cooler under my umbrella with the fan going and more than once I shared it with someone's child while in line.
The other nice thing about the umbrellas is they will double as rain protection. But I would also strongly suggest either taking a midday break back at your resort or scheduling a lunch ADR so that you at least get some AC if you'll be in the park all day.