First and foremost, that "lady" and I use the term loosely, is wrong! Transgender care DOES exist in the US!
I'm in Connecticut, and we definitely have it here! As a matter of fact, it is actually illegal in the school sytem to not
call someone by their preferred pronouns, not that anyone that I've heard of has ever been sent to prison for it. That "lady" was gatekeeping (throwing up imaginary roadblocks) you!
I'd look for a different doctor, or at least refuse to talk to that particular "lady" (secretary?) if I was you.
Have you tried going to a Planned Parenthood clinic? Many of them are known for not only giving HRT prescriptions, but can help you find a therapist that can work on a sliding fee scale. Also, check into your local LGBTQIA+ center, they might have info that could help you. I don't know about the letter recommendation in Texas, but at least these places should be able to get you started on what you'll need.
Don't give up, there's help out there, despite this backwards political climate!
And congrats for being on HRT!
Ryuichi