I was diagnosed with melanoma 1 day before heading to a planned vacation to WDW 5 years ago - I was devastated and like you, worried about not only being exposed to sun while on my vacation but even more concerned for my very pale daughter.
I actually found it very doable! First of all, we never travel to Florida in the summer when the UV rays are strongest. We generally go in the winter months - realizing that they are still there, but at least not peaked!
We started out in the room with the thick lotion type, but carried the spray on stuff for the parks and reapplied every two hours - if we were sweaty, we did it every hour - I found this to be easiest and most convenient and it worked perfectly (we came home white as could be lol)
I also found that I could walk on one side of the "street" where there was shade from buildings to help - plus we were in and out of indoor attractions so often, I really didn't find it to be much of an issue.
The boats have indoor seating and the bus stops I have been at all were shade covered as well.