No, you do not need to.
Any changes to be made to the birth records and birth certificates are called amendments to the Birth Record. After adoption has been adjudged, amendments need to be made to the Birth Record of the child which will require an issue of a new Birth Certificate to the newly adopted child.
Before annexation of Crimea and military action in Donetsk and Lugansk regions, adoptive families had to travel to the city/village/town where the adopted child was born, to receive any amendments to the birth record,. There they had to implement such amendments and receive a new Birth Certificate. The Cabinet of Ministries of Ukraine has established new procedures as for amendments of any vital records (Birth, Marriage, Death records) of the Vital Statistics offices, located in Crimean peninsula, Donetsk and Lugansk region. Such amendments can be performed in Kiev office of Vital Statistics.