We did exactly that trip last May. Here's the route and the reasons.
Travel Day 1 - I95-I287 over the Tappan Zee down 287 to NetCong NJ to stop for a couple of days and visit relatives. Avoid the GWB and the Cross Bronx Expressway at all costs.
Travel Day 2 - I80 to I81 through Harrisburg to Winchester VA where we picked up US 50 to US 17. Reconnect with I95 and stopped at the Fredricksburg KOA for the night. We did this to avoid Baltimore/DC and the no propane bridges around there.
Travel Day 3 - I95 to I295 to I64 to Norfolk. You can take 664 around Norfolk or just stay on 64 to go through. You'll need to stop on either road and prove to a state environmental agent that your propane is turned off. From Norfolk we picked up 168/158 down the Currituck Peninsula then across the bridge onto the OBX south on Rt 12. We picked up a rental car in Kitty Hawk and continued 35 miles further south to Rodanthe. We stayed in the Hatteras KOA.
Just a note. No cell service on Sprint at Hatteras. The park WiFi was poor and our hotspot didn't work either. in fact, the radio in my old laptop wouldn't even work. I suspect it was interference due to the large cell tower close to the CG.