That's tough. There is the big First UU in Austin, pretty traditional service but very political. Not sure it would fit. There is a link to it in the other thread. I would at least visit before ruling it out. I would also suggest trying the Quakers (Society of Friends), you can be a non-trinitarian for sure but of course it is a silent Quaker service.
Friends Meeting of Austin
There are some very liberal methodist churches like Trinity but there would most likely be an assumption of belief in the trinity but they are very welcoming and I do not think that there is a quiz on details of doctrine:
Trinity United Methodist Church, Austin, Texas - Home Page
I do not know of any UUA churches in Austin. St. David's though is a big diverse welcoming Episcopalian Church but they are going to be big on the Trinity.
Welcome to St. David's Episcopal Church
Short of UU, I do not think that you will find what you are looking for. Christian tends to equal trinitarian.
Good luck!
Here's First UU Austin, if this does not suit, you are most likely out of luck.
First Unitarian Universalist Church of Austin