I too start feeling ill when the sun is really beating down and I think I find that most in the World Showcase, as we're most likely to be there mid-afternoon. Even with tons of sunscreen and a hat, I still feel a bit icky when the sun pounds down, and I'm sure a lot of it is in my head, but I feel like the umbrella might block that a bit.
In 2014, while watching FoF in the Magic Kingdom, my phone said the temp was 97, but the real feel was 112, and I was standing there, and I clearly remember watching Snow White skipping towards me, then suddenly, she seemed to turn upside down and right back upwards.....then I felt as if everything I'd eaten for the last 72 hours was about to come right back up and pouring out of me, I said to DP, "I need to get out of here," and ducked into a store where I guzzled a bottle of water (I was already pretty well hydrated though), and felt a *bit* better. Afterwards, we worked our way over to Small World, where that 16 minute AC-ed boat ride got me about 95% back to normal, but yeah, the sun can do a number on me.....