I moved from New York to another State. After submission, New York adjusted my return. How do I refile in my new State with the adjustments New York made?

You can amend your return by visiting the Other Tax Situations menu in the Federal section of TurboTax. Look for Other Tax Forms in the list and choose Amend a Return. You will start in the federal section and make any changes necessary then proceed to the state section. You should make sure your returns are processed and you have received your refund, if applicable, before you amend your return, as the taxing authorities may make changes that you need to incorporate into your amended return. 

Once the amended return is complete, you will typically have a refund or tax due that is in addition to your original refund or tax due. If your tax didn't change you may not need to file your amended return, depending on the significance of your adjustments.