I changed from a feminine to androgynous name, so I didn't have to worry about whether I was transitioned. I could see waiting for the physical transition to start if it's masculine, but if it's androgynous I don't see the harm in changing it immediately.
Even with the neutral name, I gave it a year-long test run with family and friends to see if it felt right, and I was still committed, before doing the legal change. Unless you want it now very badly, a trial period might be worth considering. It could also earn you Responsibility Points with family and doctors.
Re: Driver Licenses. I got a fresh new one (for $10 or so I think?) the day of the change. New expire date. I'm thinking if my name is legally different, I should make sure the critical IDs and paperwork are up-to-date. (I'm from Washington State, you might want to check your own state's rules.)
These are just my own opinions, though. Of course, do whatever feels right for you.