Full faith and credit applies to one state accepting the judgment of another. California can't reject Nevada marriages or divorces.
But it doesn't apply to the feds, and it doesn't even apply to all state actions. I have a driver's license and a concealed carry permit. By compact, the states all recognize each other's driver licenses. They don't all recognize each other's concealed carry permits, because there is no all-state compact to do so. And within a state, my permit means nothing the moment I walk onto federal property (like the post office), unless the feds have carved out an exception (like they did for, say, the Blue Ridge Parkway - if you can carry a gun in the state in which you're located, it's okay to have one on the BRP, though not necessarily in all associated parks and facilities).