You are misinterpreting what is meant by the Full Faith and Credit Clause. The SCOTUS has already explained why a resident of one State cannot move to another and compel that State to accept laws from another.<quoted text>
Full Faith and Credit shall be given in each State to the public Acts,
Records and judicial Proceedings of every other State.
This has held true for Concealed Carry Permits, Nursing and Medical Licenses, Drivers Licenses, Insurance Licenses, Car Insurance.....
I could go on for pages, but why bother?
Just because something is legal in one State doesn't mean you can do it in another. The is the entire foundation of our Republic, States are in a sense individual countries.