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In the year you moved you will allocate the expenses for the time spent in each state for that year.
https://turbotax.intuit.com/tax-tips/state-taxes/multiple-states-figuring-whats-owed-when-you-live-a...
https://ttlc.intuit.com/community/income/help/how-do-i-allocate-split-income-for-a-part-year-state-r...
When you go to sell the house in NJ you will have to pay tax in the state of New Jersey.
According to the NJ Department of State, non-residents are required to file a tax return if they have "income or gain from New Jersey property.
https://www.state.nj.us/treasury/taxation/documents/pdf/guides/Buying-or-Selling-a-Home-in-New-Jerse...
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