Ok, first of all, there's no reason to think your federal taxes are wrong, unless there is additional information you didn't tell us. You may have just erred in allocating your income between NY and NJ.
Then, as a general rule, you owe a full year resident income tax return to your home state (your permanent residence) that reports and pays tax on all your world-wide income. That would be New Jersey. You file a non-resident tax return with New York that reports your New York-source income (income earned while physically living in or working in NY, regardless of where your permanent residence is). New Jersey should give you a credit for taxes you paid in New York on the same income.
However, you have to allocate your income between the two states, Turbotax won't assume. For example, bank interest, investments, and so on, would only be reported on your NJ return, because it's not NY-source. If all your work days are in NY, then all your wage income would be allocated to NY.
What 1099s do you have? If that's for investments, that's NJ income only. If it's for self-employment, then you allocate it to NY or NJ depending on how many days of the year you physically did that work in both places. For example, if you have a W-2 job M-F in NY, and a side gig on the weekend that issues a 1099, then your W-2 job is sourced to NY, but the side gig is sourced to wherever you actually performed the work (and it might be split if you work in both states).
If you originally owed a large sum to NY and got a large refund from NJ, that sounds like you had NJ tax withholding from your job, but you allocated all your income to NY. Was that a correct allocation? Maybe NJ says the allocation was wrong, and you owed less tax to NY and more to NJ. If that's the case, you would amend your NY return to claim a refund. However, NJ might be incorrect in their assessment.
The first thing you need is clarity on what NJ disputes. Do they dispute the allocation, the out of state credit, or what? You also need to review your return and see how you actually allocated your income.
