You must be visiting friends or family in Havelock, right? Near there, you could tour the colonial-era Tryon Palace in New Bern (home of Pepsi-Cola, by the way). Definitely visit Beaufort (NC) for the Old Burying Ground and Maritime Museum on the waterfront. (Try Beaufort Grocery or Clawson's for lunch or dinner.) Boats run people from there over to the Cape Lookout lighthouse.
Morehead City is nearby, with waterfront restaurants and the beaches on Bogue Banks (Atlantic Beach to Emerald Isle.) Fort Macon on the banks is fun to visit.
If I were you, I would also take the Cedar Island Ferry over to Ocracoke and spend a few nights on the Outer Banks. Ocracoke is very charming and there's a National Park Service campground (no hookups) by the sand dunes. You could even run up to Cape Hatteras (leave the RV in the CG) for the day via the free 30 minute ferry from Ocracoke and back. On Hatteras, see the famous lighthouse and visit the Graveyard of the Atlantic Museum.
Back on the mainland from the Banks, heading south - besides Charleston, I'd go to Beaufort (SC) and visit the small city's waterfront and then stay a couple of days at Hunting Island State Park by the ocean. (Reserve immediately since it's SC's most popular SP.)