October can be hit or miss actually. I find early october to be very humid and still hot days but it cools off (and with some luck the humidity drops too) evenings. You are doing the one thing I would advise which is come a different time of the year if that's possible... I love late feb through late march as it's spring like weather and hotter but it's not humid yet. Nov and December are nice also but it can still get a little up there but it's not summer..
To answer your question, I use a product called
Frosty Towels and disney does sell them but they are disney price. You can get 3 for the price of 1 by getting them from the website and the smaller one (wash cloth) fits really well, even in a pocket if you don't want to carry things around.. I find 1 towel per person lasts about a week but you'll want to lightly rinse it off when it starts to look less white. You do need to be careful it uses a combo of citrus acid and oils to cool you off and help you stay less sweaty but you will feel anywhere your starting to get a little sun burnt.
Suntan lotion I actually use a product that I really like which also cools you off. "Banana Boat Sport Performance Coolzone" is the full name of the product. Available just about anywhere but also from
amazon (sometimes hard to find in the large selection). It actually cools you off when you apply it which is also handy to remember to reapply it. Doesn't last too long but you get a moment of relief. I also find bugs tend to stay away and other products I've tried they seem to be attracted to it. My daughter claims it's the aloe in it (so also helps sooth starting to get burned) but who knows, never looked it up....
The biggest thing I've learned since moving here is that nobody gets used to this weather, I don't care what they say but the weather isn't all year, it's only a few months really that it's like this. What you do get used to is how to stay out of it or when it's time to not be in it. It sounds like that means don't be in a park but you need to listen to your body. If you're slowing down, starting to drag, sweaty, not really hungry when you should be.. It's time to refresh and get cooled off and after awhile you figure out when that is needed or how to stay refreshed before it reaches that far. Window shop from inside the air conditioned gift shop a little more, sit down at a counter service place inside with some free ice water. Walk through buildings you know are air conditioned instead of taking the route you don't get into the cold. Ride somethign or watch a show that takes more than a few minutes or get into a long standby line that is all indoors and in air conditioning. Then you take your break and get a ride while your cooling off anyway.
And then there is the hard fact that sometimes it's just soul sucking hot out and there is nothing you can do but avoid it every chance you can so the full circle answer is back to come a different time of year...