In Washington, both the forms for a name change petition and order and the court rules governing the procedure can vary by county. The general rules of court for this type of proceeding can be found here, including the rules governing motion practice. Generally speaking, in this type of situation you would follow proper procedure to bring a motion before the court that issued the order asking that it amend its order to include the information required by New York. While the information is not required to be in a Washington order approving a name change, I did not see a prohibition on its inclusion -- but there are general provisions that are intended to keep people from abusing the information that is included in court records, for example by using the details in a court record to commit identity theft, and I can't rule out the possibility that your local court will be concerned that those provisions may apply. You will also need to check the local court rules for your county (LCRLJ) to see if there are any that govern motion practice and name change procedures.

