We have travelled with and without our dogs. Our boarding kennel is great; we completely trust the operator. I also trust the sitter who stays in our home that we have begun using. We are so fortunate to have the choice.
We took a three week road trip to Texas two years ago with our seniors Achilles and Stella. Doing extensive research about lodging and activities helped result in an almost flawless trip.
I don't know the Outer Banks so can't offer specific recommendations. So for the moment I'll only offer a few general ideas.
Travelling during an off season is beneficial. Less crowded and generally lower prices.
If you need to stay at a hotel during your trip - La Quinta is our preferred chain. Every one we have stayed at has been pet friendly with no deposit, clean, reasonably priced, with an included breakfast (of varying quality).
Most Interstate highway rest areas have designated pet and non-pet areas. When we stop for a longer break we sometimes use non-pet areas especially if shade is an issue, making a point to be discreet, choosing a picnic table well away from other tables and gathering areas as well as leashing our dogs to the table. Though we have never been asked to relocate we have been ready and willing to.