The only "hills" on the Outer Banks area, islands or mainland, are bridges.
You will be begging for something other than F-L-A-T.
Except for US-70, I think all roads are about the same. If you ride on the highway on the island between Atlantic Beach and Emerald Isle, it is best to do so early - before the mass of tourists get out and about. Be watchful if you ride the shoulder on the highway on the island. Guaranteed flat(s) from the usual debris; except I think it is worse because there is always some construction and thus that kind of debris (e.g., nails, screws) on the shoulders.
There may be a "charity ride" down that way about that time, but I can't find a web site.
Do not have a flat in the middle of nowhere if there is no wind. The mosquitos will either carry you away or drive you insane before you can change it. There is always wind on the islands.
'70 Leopard (Swiss), '70 Raleigh Super Course, '74 Raleigh Super Course II, '73 Raleigh Grand Prix, Schwinn Montague folder, C-dale M-300, C-dale M-400, C-dale R-900, C-dale CAAD-3
I grew up in the Morehead area and left 30 years ago for a number of reasons, overcrowding and traffic being two of them. That being said, I still have family and friends in the area so I visit often and always bring my bike.
The main tourists roads will be apparent - Hwy 70 through town and on into Beaufort, the bridge over to Atlantic Beach, and Hwy 58 down the island to Emerald Isle. They are overcrowded, poorly built, and tourists are not known for being bike-friendly. I also do a lot of riding in Nags Head/KDH/Kitty Hawk and Wrightsville Beach, so it seems to be a constant factor.
There are places to ride, though. If you don't mind the fear factor of riding in traffic, the bridge over to Atlantic Beach is a spectacular ride. Once in the town of Atlantic Beach, turn left and take the road to Fort Macon. That's a Park Service road and it's patrolled so the traffic tends to be more polite. The fort is fun to walk around and they have water and bathrooms (always in short supply at a beach). If you turn right and take the road south, there are sections of bike lane along Hwy 58 but they tend to end at random moments. Traffic is REALLY heavy along Hwy 58 between Atlantic Beach and Salter Path.
Take your bike over to Beaufort and ride around town. If you want a longer ride, ride on Hwy 70 out of town toward Harkers Island - you can pick up Hwy 101 toward Havelock which is an older two-lane blacktop but with not much traffic except during rush hours. It runs out through some beautiful farming country and along the inlet and over the Intracoastal Waterway. Or keep on Hwy 70 and as soon as you are out of town, it's very pretty and takes you over some nice bridges. The Velo East Bike Club sponsors rides out that way pretty often, check their website - I think it's ECVELO.org or something like that.
In Morehead, a loop I often do is is to take N 20th Street to Crab Point Village, take Country Club Road back to Arendall Street (Hwy 70) and then back into town. I think that's about 15 miles on city streets. My brother used to live in Crab Point so I know the area. Yeah, there's traffic. You can find all these routs on Google maps. And, also, the roads that run parallel to Arendall Street in Morehead tend to be pretty with nice old houses, and the city has blocked these roads at some intersections to keep tourists from using them as cut-throughs. Keeps the traffic to a minimum.