What time of year are you going? Summer? I live in NC and visit both areas unite often. In OBX I would recommend staying near Hatteras or Duck if you like it quiet. If you like having a bunch of non beach activities, then stay around Nags Head/ kitty Hawk. We prefer the latter and highly recommend going to jockeys ridge state park. If you have kids, the aquarium and the Island farm are really fun.
For Asheville, i’d Recommend staying near Pack Square if you’re looking to stay in town. Tons of really nice B&Bs around there. If you want to be closer to hiking trails, I recommend staying closer to the outskirts near Pisgah forest. If you go in the summer, don’t miss Sliding Rock- great fun and a place to cool off. Hope that helps!