Finding places to stop along the way (during the day)

Calling ahead helps too. You're not the first people to ask. A little polite southern charm can work wonders.

When visiting the Biltmore, they put us in the overflow / ad hoc RV section.
When visiting Longwood Gardens, they gave us our choice of the far end of the lot which had trees for shade.

On both occasions we got there early so no one was in our way.