well, the first thing I would consider is re-arrange your local friend group, because they will be much better at this than I for what you can do locally. Quick google shows a few restaurants, bars, lodging providers, therapists, gyms, etc. are openly supportive and accepting. I would look those places up and first, support them with your dollars, and second get to know the staff and owners and try and get into their community. Most people who are openly supportive are eager to meet more openly supportive people and you can actually just strike up the conversation about what are the comings and goings on around JC, what else can we do, are there any organizations, etc. Often times there are support groups, get togethers, etc. This may not yield exactly what you want but it gets you plugged in and there's no better place to do that than in your own back yard.
As for organizations to support - here are the Texas orgs I would consider (some of which I am intimately familar with but I'd rather not say my involvement publicly I can answer questions privately though):
* Equality Texas - this is the premier Texas LGBTQ+ organization in the state. They are the group that ran trans ads during Abbott's state of the state address. This organization is well funded and focuses exclusively on LGBTQ+ issues, and let's face it right now that's almost exclusively TQ issues.
* Texas Freedom Network - this organization is a fierce advocate at the Texas Capitol and does a fantastic job energizing Texas youth to get out the vote (not good enough mind you, but one of the best). TFN is unique in that they focus on religious freedom (freedom from religion mostly), public education (one of the lone progressive voices at the SBOE hearings), as well as LGBTQ+ rights and reproductive rights. A small organization, but a great one with a unique cluster of issues they fight for.
* Out Youth - this organization focuses on LGBTQ+ youth in their journey, I don't know much about them but one year we volunteered to do light handyman and clean up work in one of their houses. They have a community center approach but with a traditional home where youth can come and find a safe place. we donated a xbox that summer and scrubbed the snot out of their house, painted, repaired, etc. It was a good time and it's a more granular approach to helping.
* Lifeworks - this organization is more broadly focused on ending youth homelessness in Travis County, but that necessarily means LGBTQ+ kids and young adults. LifeWorks is about to break ground in the coming months on a new 120 unit housing complex dedicated solely to helping homeless and at risk youth transition to adulthood with dignity and opportunity. This organization also offers "wrap around" services including job training, education support, trauma counseling, peer support, life skills, housing assistance, help setting up apartments with all the goods and wares needed, etc. A real thought leader in the national movement to end youth homelessness.
* Kind Clinic - this is one of the last standing organizations providing trans and LGBTQ+ healthcare without fear (well maybe some fear). I don't know about volunteer opportunities, but they do accept financial donations and contributions.
* Texas Observer - I've mentioned them anytime someone posts their articles. One of the only progressive, long-form, investigative journalism magazines focused solely on Texas. They accept donations starting at $5/month and their print magazine is a steal of a subscription. They also accept larger donations. If for no other reason, supporting the Observer to get the magazine is a great way to stay abreast of unheard of and unreported travesties in Texas (and in some case hope).
Nationally, Human Rights Campaign (they have a local Austin committee that is active), GLAAD, Lambda Legal, ACLU, Planned Parenthood (also in Texas), could all use financial support. But for being active locally, those are the organizations I know about. I also hate to say this, but trans specific organizations are not very effective imo. I think the organizations that are focused on more than just T issues do better.