MK: Tom Sawyer Island is probably the best since there's no time limit on how long you can stay there, although it is outdoors there are plenty of places that are shaded, lots of places to sit and relax/ Attraction wise would be peoplemover, HoP, Country Bears, Tiki Room, all do have a time limit, obviously.
Epcot: Living seas with Nemo, the American Adventure pavilion, Land pavilion, all are nice, chill in the a/c for a while
DHS: Launch Bay, Walt Disney Presents, again nice, a/c no time limit
AK: Rafiki's Planet Watch (if open) for indoor a/c. AK has a lot of trails with nice shaded areas, especially around the tree of life
As far as nearby spots, Contemporary would be my go to for MK. Beach Club or Boardwalk for Epcot and DHS, easy boat ride away, AK is trickier as it's separated the most but a short bus ride will get you to AKL, that's a great resort to explore on itself with plenty of activities that you can do even if you are not staying there