I live near Washington DC. When our daughter went to college in Newport News we often used the Henry Nice Bridge (Rt 301 S) from MD into VA. I think this is a good solution - maybe best solution - for you.
Be aware that it is a very tall bridge and only two lanes (one south, one north) so it is a bit more of a gut check than some bridges. More so when a truck is coming the other way. But the lanes are wider than a normal lane so there is more than enough room. Also there is a toll stop, not a big deal if you have a EZ Pass.
Second best is to take 50 to 301, to 4 to I-95. But you want to time DC very early in the morning. Rush hour here runs from about 0600 until 1000. Then they start on road repair or construction. You do not want to be between Washington DC and Richmond between 3pm and 8pm on I 95 south.
I-81 is very far west. If you can fit in a camping trip to in the Smokies or Blue Ridge it is worth the extra time but other wise stay east.
Santee Cooper State Park SC is a nice place to overnight or spend a day. Reserve ahead of time.
I also like the KOA Jacksonville North/St Mary's. Sites are a little close together but the people are very nice and the price is reasonable.
Jacksonville now has the eastern loop of its beltway complete. Much easier to navigate than I-95 through Jacksonville.