Share with us what part of the country you are moving to, maybe others are in the area and will know a good vet for you.
I would have a good vet visit before the move and stock up on anything you need, then when you are in your new home, visit dog parks or dog specialty stores and talk to people about their vets. You are bound to find others who are very particular about their dogs care!
If you can't find a vet that knows about basenjis, you will have to shop around for vets that are open minded and willing to listen to you. There is a lot of good information out there about basenjis and their health issues, so it is easy to print it and bring it with you. And of course, your dog will either like the vet or not, I never use a vet my dog doesn't like!
You may have to settle for a good or just "OK" vet, depending on where you are moving to. I would check out emergency facilities too, mine never get injured between 8-6, M-F, rather always on Sunday afternoons!
Good luck with your move!
Anne in Tampa