Yes. Since the adoption deed is validly registered and is not set aside by any court as yet its bound to be given effect to.
So you can file an application before the court of ward which can summon both parents to record their statements.
However before the court they can dispute the adoption.
So first make them agree for this and then go for court order.
Without court application also your adoption is very much valid. To remove the anomaly of names mentioned in the birth certificate you can produce adoption deed. This is very much a valid procedure.