I do not know about Austin, but in Houston, Texas, there is Phyllis Frye, who transitioned in the late 1970s and who practices law in Houston. She is also now a judge in Houston as well. She routinely gets name and gender marker changes done so I know it can be done.
If you cannot find someone in Austin, consider calling her office and speaking to a legal assistant. They may either be able to point you to someone in Austin or even handle your case for you.
http😕/www.liberatinglaw.com/
Phyllis has been an activist on behalf of transgender legal issues her entire adult life and is a very supportive and helpful human being. I will be using her for my own legal transition process later this year and have already spoken with their office.
P.S. Please note that these gender marker changes apply to things like driver's licenses, etc. If you were born in another state, that state may have SRS requirements for changing the birth certificate. I was born in West Virginia, for example, which does require SRS to allow a gender marker change on the birth certificate.