You are welcome to join whatever political party you choose. Being trans doesn't mean you must become a left winger. However, there is plenty of truth in saying that the current GOP isn't as friendly to the lgbt community as the current democratic party. You may choose to vote Republican based on other reasons, but they aren't necessarily adoring of our community. Look at the support for gay marriage, enda or anything else lgbt oriented. The dems are generally more supporting of lgbt issues. Still, there are many different people in each party and it's best to vote for who is most representative of your views and ideas. And, of course, the dems have their own bigoted history when it comes to the lgbt community. As it stands now, they seem to be the more sympathetic, but that doesn't mean you have to practice single issue voting. Your political beliefs are your right. No one should tell you who you should vote for; that is up for you to decide.
One thing I will say is that Ted Cruz doesn't favor the Lgbt community based on his past statements and actions. I know someone will drag up an innocent looking quote which attempts to negate everything he has said or has done, but that doesn't excuse everything in his past or justify every statement he's made. I can understand people voting for the Republican party or Cruz for many reasons, but support for the lgbt community is not one of them. Well, at least for now. Things can change as they have in past and parties shifted positions and priorities. As it stands, the more inclusive party for the lgbt community is the democratic party. However, if you have more important factors at hand when voting and you tend to lean right, then you have made up your own mind and should not worry. There is no reason you should have to vote for the democrats just because you're trans. I myself hardly go to the voting booth with lgbt issues in my mind; even if they are a factor in who I'll support, it is only one small element. So you will be no different than me or any one else that doesn't practice single issue voting. Personally, I don't understand voting on one factor alone unless it is urgent.