Full Faith and Credit clause does not require state #2 to obey every order issued by state #1 - state #2 is always able to re-examine the validity of such issues as personal jurisdiction over the defendant(s), which is clearly absent here. Lots of cases establish this principle.
If Obama submitted a declaration of cadidacy in the State of Washington, he subjected himself to the jurisdiction of the State of Washington for purposes related to the election. Therefore, there is personal, subject matter and venue jurisdiction.