I also wanted to add that if walking around the parks is also going to be too much sun, try the route throught the stores - the ones along Main Street at MK connect a lot in between, especially on the Emporium side. We also like walking through the shaded areas beteen Innoventions at EPCOT, but it was hard to find a shady route through World Showcase. MGM shop areas also connected in some areas. The air conditioning is also very, very nice, especially when we go in August.
Actually, we go in August because of my need to stay out of the sun. The parks are open well into the night, and we take a long break during the afternoon to avoid the heat and sun. Also, it rains a lot in August (those great afternoon thunderstorms), which blocks some of the sun (not all of it though - I still gob myself up with SPF45 sunscreen)