The short answer is, New York should eventually figure it out and send the extra back, assuming that your amended NY return matches the corrected return they created.
The medium answer is, you can call NYS Tax to talk to them. In the past they have been very reasonable to deal with over the phone, much better than the IRS.
The long answer is, more detail is needed and why did you change anything in the first place? A change to New York would not necessarily change your federal return. It might if NY found a mistake before the IRS did -- for example, if you forgot to include a 1099 for some kind of income, and NY found it first, then you need to amend your federal return to report that income. But if it was a NY specific change, like to an NY deduction or credit, that might not have affected the federal return at all.
Then, after preparing the amended federal and state return, you should have stopped before filing and made a careful comparison of the amended NY return to the NY return as changed by NYS Tax. If the amended return did match the changed return, you should not have filed the amended NY return, only the federal return, because NY already had the change on file. If the amended NY return did not match the changed NYS Tax version, you should have stopped and investigated why your amendment did not result in the same change.
Your comment "TurboTax must have told NY I owed them $2,000" shows you have not really analyzed and read your tax returns. You need to print out your original returns (which you should have saved) and the amended returns, and compare them carefully.
What happened was, when you prepared the amended NYS return, Turbotax assumed you had received the full $3000 refund you originally claimed. Since the new tax return called for a refund of $1000, you needed to pay back the extra $2000. This is the correct calculation, except that you did not receive the original $3000 and Turbotax did not know this. See lines 78, 78a and 79 on form IT-201-X (your amended return).
At this point, NY should figure it out eventually and send you back the $2000 automatically. Or you can call them.
