Would you prefer the coastal route (US 17) or the mountain route on The Skyline Drive and Blue Ridge Parkway?
You should be able to do either with about 9 hours of driving time not counting stops.
I don't blame you for not wanting to drive I-95.

How long do you plan to spend making the drive (which is probably 5-6 hrs. max directly via I-95 and I-85) and what time of year are you doing this?

Lived just north of DC, in Richmond, and in Piedmont NC the first half of my life and still travel in the area sometimes; I can't say I find much scenic about the direct route. (Which is not to say there aren't a quite a few miles of pretty wooded areas here and there but nothing outstanding.) But if you want to take a little time and go out of your way - maybe considerably - you could do it either via the mountains or the coast, at least part of the way.

It's a quick drive on I-66 from Washington to the Skyline Drive, then slower on SD through Shenandoah National Park and to Charlottesville. If you want to stay in the mountains and on the Blue Ridge Parkway, you're going farther and farther west of Raleigh. (Skyline Drive and BR Pkwy. are slow-speed roads, so you'll take at least twice as long as on an interstate, even if you don't stop. And I can't imagine how anyone could drive there and not stop to admire some of the views.) But you could cut over from the parkway to I-40 and make it to Raleigh from Winston-Salem in 2 hrs. or less. (That part won't be scenic.)

Or you could drive to Williamsburg/Jamestown on I-64 either from the mountains (maybe 3 hrs. from Charlottesville) or I-95 in Richmond (less than an hour off I-95), drive the Colonial National Historic Parkway and stop in Colonial Williamsburg and/or Jamestown. At that point, your best bet for more scenic driving will first take you through some congested areas around Newport News/Norfolk before you can get to a road that links you to the Outer Banks in NC. (Someone else will have to tell you more about the route from Williamsburg to the Outer Banks since I haven't been there for 30 years.) From the Outer Banks the most direct way to Raleigh is on 64, which is a pretty good road but not much of it is limited access.

If you don't plan to spend at least 2 or 3 days doing this 5 hr. trip (stopping in the nat'l park, Monticello, Williamsburg, Kitty Hawk/Nags Head, depending on your route) I don't think I'd bother, though.

Booth cities are in the Piedmont, so there is no spectacular mountain or seacoast scenery without going pretty far off a direct route, but you can take US 15 all the way from Washington to Chapel Hill before turning east to Raleigh.

It is mostly a divided highway, though not limited access, it passes around most towns, but you have the chance to go through towns when you want to, the scenery is pleasant once you are out of Northern Virginia, and there is very little traffic.

Thanks so much for this information! Being from the midwest, I am/was totally clueless about this region! We are visiting relatives in Raleigh, so will decide which we can fit in, won't be able to do completely, did not realize it would double the travel time.
Thanks!

Any good sites or places that are must see/do in Raleigh or Washington DC July 14-21?
We would love suggestions that the locals know of!