Basically you have to file part-year resident returns for both states. If you had any New York source income after you moved to New Jersey, the New York part-year resident return will cover both the resident and nonresident periods. If the sale of your New York house closed after 8/12/22 the income from the sale is New York source income while you were a nonresident.
For New Jersey you have to file two separate returns, a part-year nonresident return for the time that you lived in New York and worked in New Jersey, and a part-year resident return for the time that you lived in New Jersey.
TurboTax will guide you through this. Answer the questions about residency and income sources carefully, starting in the personal info section at the very beginning of the federal interview.
