^^The other state probably wouldn’t even know they were first cousins, because all they would be presenting at the other state, if they moved, if they were forced to present anything, is their marriage license. The license doesn’t say anything about it, the questionnaire for the license probably asked the question. Possibly, not though. Possibly, you’re just expected to know the law.
I think there are still a few states that do not accept the marriage as valid if you are first cousins married in another state. That’s the law on the books anyway, I doubt it’s enforced though. For the most part states have reciprocal agreements about marriage and acknowledge any other states’ legal marriage.
A clear, and fairly recent, example where states did not acknowledge another states marriage because of their own laws is gay marriage. Gay marriage, for years, the reciprocal agreement was not being used, gay married people were not married in their new state, if they moved to a state without gay marriage. That’s been fixed now. It was fixed at a federal level. A lot of people argued marriage has always been a state level thing, and the fed stepping in was overreaching.