I have always had a car, so I do not know about transportation from the WDW resorts....
As for HH...there are over a dozen courses within 15 minutes from the resort.
There are three courses at palmetto dunes which are right across route 278 (not easily walkable, but less than 5 minutes away).
Sea Pines (which has 3 courses, including Harbortown Golf Links - very expensive, where the PGA plays) is about 15 minutes. I played at Sea Pines (not the Harbourtown course). It was was beautifully maintained and a wonderful round. Just bring bug spray....lots of little nat type bugs that bite...but they leave you alone completely with bug spray. Beautiful courses - all near by. Enjoy!