LOL!
I sympathize with your plight, but if you'll permit me to slip on my curmudgeon-iest, most snarky persona (
in other words, understand that I'm engaging in tongue-in-cheek hyperbole here), here's my reaction:
Here me out for a second. Yes, I could rent a locker but that is a hassle.
You know what else is a hassle? A goshdarn
stroller. They're already clogging up every bus and walkway on Disney property, slamming into people's ankles and ruining their vacations -- nay, their lives. Why do it when you don't have to?
With a stroller, it can go with you everywhere. Parking it is easy, you can leave it parked where it is for a while, and you don't have to lug around that backpack with everything in it.
A stroller can't go with you everywhere, and everybody hates looking at it parked all over the place. And why in the world would want you take "everything" with you, in a backpack or otherwise? If an item doesn't fit in a tiny crossbody purse, you do not need it. Imagine a world where you don't have to spend portions of your vacation time parking or hunting for a stroller.
I know it is better to pack light. However, there is a need for a few extra things in the heat/humidity of FL. You need flip flops to put on for water rides and cool off in the fountains.
No, you don't (and unless you're a member of the cast of "Friends," stay out of the fountains).

Wear waterproof hiking sandals to the park (e.g., Tevas), and your feet are ready for anything.
You need sunscreen (at least I do because I do not want to have skin cancer surgery again).
You've got me there -- nobody wants you to risk your health, but why not bring a small, sample-sized bottle of sunscreen with you instead of the gallon jug of it from BJ's Wholesale Club (you can refill it from a bigger bottle in your hotel room)?
You need to be able to carry water.
Free iced water is available, on request, from any CS venue. I like to carry it right in my hand, no wheeled conveyances needed. I drink it, and then throw the cup away in the nearest appropriate receptacle. If you must carry an extra supply of water with you, get a small sports bottle on a carabiner and refill it from drinking fountains as needed. Or just drink from the drinking fountains -- it's what Walt would want you to do with the drinking fountains, anyway. "Drinking" is right there in the name.
Having that ice cold misting fan is nice. I can even save by having one of those backpack cooler to bring my own drinks and lunch in.
If you find you need some water in your face, ask a loved one to flick some on you, or simply bathe it in one of the said drinking fountains -- or to cool off, do the redneck version my mother taught me -- duck into the nearest restroom, remove your shirt, run cold water over it, wring it out well and put it back on (obviously this requires wearing clothing that will stay opaque when wet). As far as carrying lunch, granola bars are lightweight. So is money to buy lunch (just use the money you were planning to spend on jugs of water, misting fans, strollers, backpack coolers, shopping carts, Sherpas, etc.)
Sorry, OP -- I know I'm giving you a hard time (all in good fun), and please take what I'm saying with the humor in which it was intended. It is your life and your decision and you should not be concerned with what anyone else thinks about it. Still, I would respectfully suggest that if you're bringing so much stuff to the parks that you're considering buying a baby stroller just to tote it all around, I think you might have an overly expansive definition of what you "need" with you in the parks. I challenge you to try, for just one day, breezing around the parks with no large backpack and no stroller -- just a carefully-curated set of items (sunglasses, small sunscreen, Band-Aids, disposable poncho for water rides, phone, a granola bar) in a small crossbody purse or mini-backpack -- and see if you don't feel freer, and more able to enjoy the parks and your family, than you ever have while you were carrying around a huge burden of "stuff."