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An adult name change isn't a "fill out a form and submit it" kind of thing... It requires the person who wants the name change to fill out an petition for an adult name change and file it with the court. Then the clerk schedules an actual hearing where the person appears in court and explains to the judge why they want to change their name. As long as it isn't to avoid creditors or evade law enforcement, the judge will then approve it and enter a formal legal order for the name change. That order is then forwarded to the Dept of Vital Statistics to amend the birth certificate.
This is a link to the entire process with forms: NY Adult Name Change.
But you can't "vacate" an adoption as they are permanent...