Both the Red and White in Hatteras and Conners in Buxton have a nice selection. I don’t think they do delivery, but much closer than FL in Avon. They may not be as big. A store, but both are clean, well stocked, friendly staffed..frankly, I prefer them to FL.
When in Hatteras Village, the majority of our grocery shopping was at the (newly redone) Red and White. It easily had everything we needed for the week. Yes, it was a bit more money, but just like when I'm working, my vacation time is valuable. Not having to drive up to FL in Avon was worth it to me. We did stop at Conner's when we were in Buxton for the day shopping, but we didn't really need much, so ended up just getting firewood and snacks. Both are great alternatives to FL (Not that there's anything wrong with FL, just not as convenient when you're further south).