Martinez v. Bynum 461 US 321 involves a US citizen child living with his parents in Mexico traveled to Texas in order to attend free public school. Under Texas Education Code 21.031(d), the district is allowed to deny tuition-free admission if the child lives apart from the parent or legal guardian and is present in the district "for the primary purpose of attending the public free schools". The court upheld such a residence requirement because it "furthers the substantial state interest in assuring that services provided for the State's residents are enjoyed only by residents", and "does not violate the Equal Protection Clause of the Fourteenth Amendment nor burden the constitutional right of interstate travel". Plaintiffs argued various constitional violations, including Equal Protection, Due Process, and Privileges and Immunities, but not Full Faith and Credit. We may assume that if there were any hope of such an argument, it would have been made.
In Vlandis v. Kline 412 US 441, it was held that a state cannot permanently relegate an applicant to non-resident status for having a legal address for any time within a year of applying for admission. But the court also explicitly states:
Our holding today should in no wise be taken to mean that Connecticut
must classify the students in its university system as residents, for
purposes of tuition and fees, just because they go to school there.
Nor should our decision be construed to deny a State the right to
impose on a student, as one element in demonstrating bona fide
residence, a reasonable durational residency requirement, which can be
met while in student status. We fully recognize that a State has a
legitimate interest in protecting and preserving the quality of its
colleges and universities and the right of its own bona fide residents
to attend such institutions on a preferential tuition basis.
The Full Faith and Credit Clause states:
Full Faith and Credit shall be given in each State to the public Acts,
Records, and judicial Proceedings of every other State. And the
Congress may by general Laws prescribe the Manner in which such Acts,
Records and Proceedings shall be proved, and the Effect thereof.
This pertains to recognizing the status of evidence and rulings, between states. If New York finds that Smith is liable for a sum of money, that finding is a legal fact in all states, and Connecticut cannot say "We do not recognize the judgment of New York". The only connection between Full Faith and Credit and non-resident tuition is that both include the concept "other states".