So as I understand it, "full faith and credit" means Arizona has to accept that Hawaii's laws and judicial decisions apply in Hawaii. I don't know how far that applies to public records -- there are arguments to be made on both sides -- but Arizona can't use the clause to undercut Hawaii's account of what its own records say. The clause would tend to support Hawaii more than Arizona in this case. More here.
I think I agree with that statement as far is it is limited to effect in Hawaii, but we're now talking about those records' effect in Arizona on Arizona law.
I know it's never been tested this way, but the Article speaks to how those records would be "proved," that is, Congressional action to the manner of proving. So the Constitution did not automatically imply that a state was forced to take another state's word on the matter.
Arizona could have demanded first-hand proof by visually inspecting Hawaii's records, if only to force Congress to exercise its power to "prescribe the Manner in which such Acts, Records and Proceedings shall be proved."
-PJ