If you're moving to another state and taking a firearm with you when you move, the firearm remains in your possession, so there's no reason to go to an FFL: no transfer between two people has taken place.
You do, however, have to be sure ahead of time that you'll be complying with the laws of the state you're moving to concerning the types of firearms you're allowed to possess, registration, if that's required, and any permits required by that state.
But those things are off-topic for this thread.
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