Northeast Appalachians must be beautiful but from my southern experiences, can highly recommend anything along Blue Ridge area .....from north to south, in the Carolinas, a nice 7 day trip
Roaring Gap (Ross)
Mount Mitchell (Hawtree)
Biltmore (Ross)
Linville (Ross)
Grandfather (Maples)
Highlands (Ross)
Wade Hampton (Fazio)
There are many other courses in these areas especially around Highlands area. Of all of these, I would suspect the best vistas may be from those I have not visited which are higher or have spectacular views...
Linville Ridge
Cliffs at Glassy
Also, if you drive along the Blue Ridge parkway on a golf trip, I would suggest a rental car so that when you warp the rotors from all the braking, you can simply turn the car into the agency.
What about the Rocky Mountain trip ? Need the western input.