OK, it seems to me the logic here is that the baby was born in Arizona, and the county where he was born has his birth record, and you're amending that record. So it seems like anything you do to change that record has to be done in Arizona. Also, check everything I'm saying in this answer with a lawyer before you do anything.
If there was a paternity challenge, or a woman was trying to force a guy to acknowledge paternity and he was hostile to the attempt, then a DNA test would be required. But if both of the parents know they are the parents and simply sign something saying so, the courts won't force them to also run out and get a DNA test when they turn in the form.
Depending on the state, a voluntary acknowledgement of paternity might be an actual form (in Washington, I think it's a carbon form with different colored pages), but in many states it's just a one-page document produced by a lawyer. In Arizona it might be the latter.
You can get things notarized at a full-service bank.
Be sure you check everything I've said with an attorney in Arizona to make sure, before you try it.