I was born near Houston as well and I'm trying to deal with this from Oregon. This was awhile back that I first started dealing with it, and either the Texas records website wasn't up to date or I misread the requirements, as I missed the part about court-ordered gender change. I sent my doctor's letter, a photocopy of my driver's license and social security card (both the old ones and the ones that match my current name and presentation), a certified copy of my name-change court order, a check for 37 dollars that I could not afford at the time, the paperwork Texas told me to fill out, and some documents to verify and explain my PO Box address.
I don't know about Arizona, but in Oregon you can file to get your gender legally changed by using the name-change forms and crossing out the word "name" and writing in "gender." I've never heard of people having trouble with it, but a judge might ask some questions.
Texas took several months to respond to my request. They refused to honor it. They didn't change my name, they didn't send me a copy of my birth certificate under my old name, and they didn't refund my money. In their letter, I was told that my court order for name change was not a certified copy, that I needed to change my gender with a judge, and that I had to explain what I wanted a copy of my birth certificate
for.
I was given 2 months to respond, but I couldn't handle it then and so I'm sure I'll have to give them more money when I get brave enough to deal with it. I understand the court order part. What made it frustrating and scary was that they refused to recognize my name change with valid documents, so I feel like there might not be a right answer to their questions about what I want to do with my birth certificate. I've had enough customer service nightmares just changing internet and utilities that the idea of wrangling with state bureaucracy is dreadful.
Anyway I'll try not to rant on your thread. There are worse states to be born in. Good luck. You should be able to do this with a little persistence.