Changing an Adult's Name: Double Name Change

My question involves name change laws in the State of: Mass.

I'm in the process of getting my name legally changed (paperwork has already been submitted to the court). Next step will be to get my bc and ss card changed, and then get a state ID. This will be my first state-issued ID as I have only recently turned 18 and had no prior ID/license.

Here is my problem (*not real names):

Birth certificate has the name Jane Lisa Brown
SS card has the name Jane Lisa Doe
Legal name change is Michael John Doe

My name at birth (parents were not married so it is my mother's name) was sent to SSA before my birth certificate was issued (with my father's name). Will I be able to get my ss card changed from my birth name to my newest legal name? Or do I have to try to get it to match the birth certificate first? When I called SSA, the agent just said to bring all my documentation, but I'm not sure I have any. I did indicate the birth name on my name change petition. Will the court include that name on the order? And my mother uses my birth name on tax returns since that's the name associated with my ssn. Could I present copies of tax returns as proof?

Changing an Adult's Name: Double Name Change