I was born in TN, but I live in NJ. I'm currently in the process of changing my name in NJ Superior Court. Today I just confirmed everything with the judges secretary from TN Birth Certificate that I must bring to court to the court date. Anyway, you change your name in the state you are a resident in (usually the one your license is with) and once you have the court order for the name change you send a certified court copy to the Registrar of Vital Statistics of the state you were born in to update your birth certificate. (In my case TN.) By law, you must do this step.
For Gender it is the state you live in for DMV and for Social Security and Passport a Medical doctors letter with the appropriate wording and license#s. (I received 5 notarized copies from my doctor early this week.) Every state is different on the gender marker for birth certificate, some just a simple letter, others SRS and in my particular case, TN will NOT change the gender for ANY reason. To further add insult TN not only will not change the gender, but in the name change, you get another birth certificate with the old name crossed out (but still visible) and your new name printed above it. Thankfully once I get the driver's license and Passport change, I'd be happy if my birth certificate never has to see the light of day again.