Yes, you'll be able to use TurboTax to do your tax returns.
Since you were a resident of NJ at the start of the year, the question is whether your move to NY was a true change of domicile at the time. NJ's rule is that your domicile continues until you establish a new permanent home elsewhere. You can only have one domicile at a time.
If you did change your domicile, then you will file as a part-year resident of each of the two states.
But if you kept your domicile in NJ during your time in NY, then for tax purposes you would file as a non-resident of NY and a full-year resident of NJ. These concepts are explained in detail in this NJ tax publication:
https://www.nj.gov/treasury/taxation/pdf/pubs/tgi-ee/git6.pdf
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
