I think the answer is sometimes yes, sometimes no. U
For me, my birth certificate is in a state that STRICTLY will not change your birth gender until you've had SRS.
However, I was able to go to the DMV and get my license changed to the gender I wanted (albeit it was a long process) and I only did it because I pass well enough. Otherwise, you have to have SRS in my state to have it changed (very strict state).
However, in places like California or Massachusetts, once you change your birth certificate and all your documents, on paper, you're that gender. So, in theory, yes, you should have the same rights.
I will say; however, if you have a prejudice adoption agency and they do a background check YOUR OLD NAME WILL COME BACK TO HAUNT YOU. Unfortunately, a lot of your old information will stay on a background check and again, if that adoption agency thinks transgender is wrong, they could just reject you simply on that alone.