You say "We still own a home in the former state" (NJ). But you don't say what you intend to do with it. If you plan to split time, in the future, between the two residences, NJ could well consider you to still be a full year resident for 2022 and future years.
You should review the residency rules. https://www.state.nj.us/treasury/taxation/pdf/pubs/tgi-ee/git6.pdf
That said, I agree with zoltanb, there is no procedure for notifying the NJ tax authority of you change in residence. The fact that you file a part year resident return with a new address is sort of informal notification.
