You might try the Amistad Community Health Center and see if you qualify for subsidized health care through their facility. Unless you get some kind of insurance subsidy, the cost of the premiums will probably far exceed the cost of the prescriptions. Hormones are pretty cheap, really. Blood tests aren't cheap, but again Amistad could help with that depending on your income.
If you've been living full time for a year, many therapists will take that into consideration and provide a recommendation letter in a short period. Many therapist don't require an office visit and can work over the phone or Skype so your not limited to your area; however, the Psychology Today website lists a number therapists in Corpus with transgender tags. Look for one that offers a sliding scale. Many will do a free consultation. Explain your financial situation and your real life experience. Try offering them 1/5 of their scale. You may be able to get said letter in as little as two paid sessions. With the letter of recommendation, the health center may be willing to administer the treatment.
I think you'll find this path cheaper and easier. At least it did for me. Good luck with it, and let us know how it goes.
All the best,
--AshleyP