Welcome to RVing! One thing to remember is that you will learn along the way. We travel annually in and out of the NE annually but always prefer the 84 out to 81 route with a larger RV. There are many ways to go and many things to see, just depends on your likes and a little bit on the weather.
But with that being said, if you are truly interested in City sights then a ride down the coast can be worth it. DC has a great park "Cherry Hill" where you can either get the metro or buses near or at the Campground to go into and around DC to visit it all.
There is the Bay Bridge & Tunnel drive (there are tolls) but it's an experience in and out of VA & DE.
Then on the other side taking 81 south - you can easily get on the Sky Line Drive and visit Shenandoah NP or part of the Blue Ridge Parkway (just need to make sure your height of your RV is able to go through the tunnels) just need to do some advance research and make some calls. These drives are very relaxing 35-45 mph speed limits, great sights but you can't be in a hurry to get where your going. There are stops, walks and hikes along the way. Or make a stop in and around the "Dutch Country" of Lancaster PA.
Otherwise when you get to SC don't miss a stop in the North Myrtle Beach area (plenty to do and see) and in the Historic area of Charleston.
Hope this is a start to get you out there and enjoying. We have had warm and cold (snow) weather so it's important to watch the weather as you travel.... but all has been a fun experience, even the challenges that have happened.