Saratoga is definitely the best for walking - it's a big sprawling resort with lots of shady tree-covered sidewalks, and bridges/paths connecting the different sections, as well as long walking trails over to Disney Springs from two different sides, and long paths along the golf course and woods all the way to Old Key West. A side-perk of Saratoga, aside from the direct connection to Disney Springs, is that with a short walk to the Disney Springs boat docks (right next to the Congress Park section), you can hop a boat to Old Key West OR to Port Orleans French Quarter and Riverside resorts - a nice, relaxing boat ride gives you access to even more lovely, tree-lined and shaded walks around those two resorts. French Quarter in particular has lovely cobblestone streets and sidewalks heavily covered by trees, and a lovely little bar and jazz music spot at night - Riverside is massive with plenty of trails throughout, and a great bar, decent restaurant, and Yehaa Bob at night.
Old Key West is also large and has plenty of sidewalks for wandering, and that same interconnectedness with Saratoga Springs.
Fort Wilderness is another big, sprawling property with nice woodsy walks, trails through the woods, tons of tree cover, waterfront areas, and more.
Beach and Boardwalk have plenty of walking options - not only the bay around Beach, Yacht, and Boardwalk, but also over to Swan and Dolphin, the Fantasia Gardens minigolf course, but also a long walk over to Hollywood Studios...but there are only a few sections of the walk that have tree cover - so while there are plenty of paths to take, there aren't many with shade.
Copper Creek/Boulder Ridge as part of the Wilderness Lodge complex have nice shady grounds to walk around the pools and the cabins, through the cypress trees - though the wonderful long walking/jogging trail that connects all the way to Fort Wilderness has been closed a while as Lakeshore Lodge is being built - there are hopes it will eventually re-open allowing walking connections between all three resorts.
Polynesian would probably be next, as having lots of little trails between all the sections, with lovely tropical plants, tiki torches, and theming - and there is the ability to walk around the entire lake to Grand Floridian, Magic Kingdom, and Contemporary - though those walks are not tree-covered or shaded most of the way, so that may earn them a little demerit.
Riviera is similar - there is a lovely walk around the lake over to the sprawling Caribbean Beach resort area - the walks to and from Riviera are in the open and direct sun much of the way, but once at Caribbean Beach resort, some of the trails and paths are fairly well tree covered.
AK Lodge does have good tree cover, but the only real walks are the nature trails out back, and the sidewalk between Kidani and Jambo - it's a nice walk, but not very long or sprawling.
