Most of these suggestions are relatively close to your proposed route:
Zion & Brice - Angles Landing hike in Zion for the views
If you're in the area, Telluride has a Hot Air Balloon Festival in the beginning of June (2nd to 4th this year), we had a great time years ago. Wasn't able to go up in a Hot Air balloon though - we arrived with no advanced plans in place, ended out driving to Snowmass and lifted off there.
Durango - If you've never done it before I'd suggest carving out a day for the Durango Silverton Train ride. Truly spectacular.
Nashville can supply some fun evening entertainment (we prefer to reside in the smaller, less commercial bars during out stays). Pretty much everyone showcases talented musicians.
West Virginia - If you fancy a thrill (and are in decent physical condition), you may really enjoy Seneca Rocks - It's a rejuvenating experience, especially when you get to the top and the wind's blowing!
Northern Virginia & Maryland - from the South East, Blue Ridge Parkway runs into Skyline Drive - both offer great scenery (especially Skyline drive).
From your Map, Looks like you'll be driving through the Outer Banks...Good choice (make reservations early)
A short detour from that area; If you have interest in Museums, visiting the DC Monuments, and spending time in a Sailing Town (with a drinking problem) I'll gladly extend an invitation to a driveway in historic Annapolis (35 minutes to DC and all it has to offer)...most people don't make it out of Historic Annapolis! Gladly throw in the use of two e-bikes to get you around the area.
Looks like a very fun trip!