How long are you planning on touring "around a bit" before returning to North Carolina?
As
Spdracr39 has said, you will want to register your motorhome in North Carolina if that's where you reside (hold your driver license and consider domicile).
The 15 day rule that Arizona says a vehicle must be registered by is for their
own residents or people moving to AZ. It is not meant for those who are from another state and purchasing a vehicle there to ultimately register in that other state so I think you can disregard that 15-day law.
I've purchased vehicles and RVs in another state several times and usually am back to either my home state or as in our case (we used a Montana LLC during our full-time years), was able to get the vehicles/RVs registered in less than two weeks. So what we did was to snag a temporary tag from the state where we purchased the vehicle to allow (kind-of-sort -of) us to travel to where we needed to go or to get the paper work completed (when we were full-time).
I say "kind-of-sort-of" as we've been told that technically, that a temporary tag is considered legal only in the state that issues it but one of our lawyers said that it could be argued that since it's a legal issuance that it can be used out of that state for legal mobility to the home state. I can't be sure if this is an absolute correct interpretation or not but we trusted this particular attorney as he had helped us with a lot of other legal issues pertaining to vehicle registration and his implication was "don't worry about it."
So with that said, I may start at that point and research Arizona's temporary tag offerings. I see that they do have a 30-day temporary registration but you'd have to check into it further to see if it pertains only to their own residents or if it's available for out of state visitors purchasing a vehicle in AZ.
AZ MVD -- 30-Day General Use Permit
Ideally, I'd check into seeing if North Carolina will issue a temporary tag and send it to you in AZ and if they could overnight it to you the day of the purchase so you'd have it the next day or day after. I'm not sure if that's even possible but you might want to check on it to see.
No matter, I'd get the North Carolina paperwork for title application started as soon after the purchase as possible and as soon as you know the VIN so you can send it to them from AZ. It says on this page that you can evidently mail the application in:
North Carolina DMV - Title Requirements- If mailing in the paperwork, the individual titling the vehicle must provide with their title information a black-and-white copy of their valid North Carolina driver license.
If you plan on traveling for a month or more, you might want to arrange for a mailing address in AZ or somewhere in-between where North Carolina can send you your plates, registration documents, etc. I believe you can do this with USPS by using general delivery to a specific post office. That way, you can avoid signing up for a mail forwarding service such as full-timers use.
USPS - General Delivery
I'm not sure if your RV needs a state inspection or an in-person VIN identification from your North Carolina DMV but if NC does require this, I suppose that could be an obstacle as you probably wouldn't be able to fully complete the registration process and obtain permanent plates if they have to physically inspect the vehicle.
So I think it's best that you make some calls both to the the NC DMV and the AZ MVD and get clarification if any of this is feasible.
But let's see what others say.
edited to add: Also, and I know it goes without saying, be sure to per-arrange for insurance and research/shop/etc. before the purchase as you'll obviously want to be sure to have it properly insured as soon as you can after the purchase and before you start to do your sightseeing.