Generally, your process is this:
1. In the Personal Info/My Info section, be sure to enter your home state as New York (what it was on 12/31/2021) and your prior state as New Jersey.
2. Enter your NJ return first. TurboTax will tell you the forms you need to file.
3. Enter your NYS return next. TurboTax will tell you the forms you need to file.
The NJ return will calculate a tax for NJ. This will be applied on your NYS return as a credit against NYS taxes.
"get a return for the property taxes I paid in NJ from Jan-April" - I am not sure if you are referring to property taxes or income taxes. We normally do not answer questions here about property taxes anyway.
NOTE: as a now NYC resident, please review your NYC withholding. It is a common problem that I see that taxpayers who live in NYC expect their NJ tax to offset their NYS tax (which it does, sort of), but since some taxpayers I have talked to didn't withhold NYC tax for whatever reason, they ended up owing money on their NYS return (NYS collects NYC tax).
This may be worth a call to your payroll department to make sure that your NYC tax is being withheld.
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