Hi martintaki, thank you for joining the "Ask the Experts: Moving" special forum today.
Questions regarding Form 1040NR is out of scope. However, I am happy to relate to your questions in general.
Unless you are military, the moving expense deduction has been suspended for tax years 2018 through 2025 as a result of the Tax Cuts and Jobs Act that was signed into law in late December of 2017.
In general, when you move from one state to another state or overseas, you do not need to notify state/local agencies, you show intent by evidence of "leaving the Domicile", such as absence from the prior location, cancel driver license, voter registration, insurance, sell your main home. Below links are for more information.
What is my state residency status?
https://ttlc.intuit.com/community/state-taxes/help/what-is-my-state-residency-status/00/26000
What is Your Domicile?
https://www.nolo.com/legal-encyclopedia/what-is-your-domicile.html
Please also refer to IRS Pub 4588 Basic Tax for Green Card Holders Guide, Understanding your U.S. Tax Obligations - "What if I surrender my green card?"
https://www.irs.gov/pub/irs-pdf/p4588.pdf
Hope the above helps.
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