2021 Taxes - NJ claiming underpayment

I filed my 2021 taxes via TurboTax Premier. There were no issues raised via the pre-e-filing TurboTax review. I was expecting a federal refund of approx. $3,300 (received) and a NJ refund of approx. $850. I have now received an underpayment notice from NJ for approx. $4,200 + penalty + interest.

 

I believe I understand the issue. I live in NJ and have 2 x 2021 W-2's, one from a job in NJ and one from NY. Both W-2's were automatically uploaded into TurboTax: one shows the state as NJ, the other as NY. However, TurboTax only prepared a NJ return + the federal and nothing else. I assumed everything was OK and that NJ would see my NY W-2, but the 1040 (NJ part) doesn't show my NY income and NY state tax paid (+ city taxes). The underpayment claim from NJ almost exactly matches the NY state tax I paid per my NY W-2. The error may be on my side in preparing the return, and surprised TurboTax didn't catch it.

 

Is this the correct path forward:

 

(1) File a NY nonresident income tax return

(2) File a Schedule NJ-COJ (Credit for Income or Wages Paid to Other Jurisdictions)

(3) File a NJ-1040X (NJ Amended Income Tax Return)

 

Can I do all of this via TurboTax, or should I work directly with NY and NJ tax offices?

 

Thanks in advance for any help / suggestions.