That is ridiculous! Fist of all, it would mean that some people would have to travel out of state to get married. That in itself would violate the equal protection clause of the constitution
Secondly Full Faith and Credit does not apply in all circumstances. For instance, gun laws are not transferrable across state lines. Before the Obergefell ruling, same sex marriages were not recognized by states where it was not yet legal
In Loving V Virginia, that interracial couple got married in a state where it was legal, moved to Virginia and were thrown in jail
Face up to it. The Articles of Confederation were replaced by the Constitution and the Civil War, the states are bound by the Bill of Rights