I'm gay and got married in another state, can I now get divorced in Texas?

The short answer is yes.  Even prior to the U.S. Supreme Court's decision to allow gay and lesbian couples to marry, the Texas Supreme Court heard the case of State of Texas v. Naylor.  In that case, the highest state court decided that the Texas Attorney General could not halt the divorce of a same-sex couple who had been previously married in Massachusetts.  A Travis County judge divorced the parties, and the Attorney General appealed that divorce.  The Texas Supreme Court ruled that the Attorney General essentially did not file their objection to the divorce timely, and the divorce was upheld.  Shortly after that ruling came down, the U.S. Supreme Court issued their ruling in Obergefell, ET. AL, v. Hodges, which allows same-sex couples to marry.  It will now be very difficult for a judge to deny the granting of a same-sex divorce in Texas.  Although, the county of filing may play a role in a potential delay,  (ie; conservative counties may create bureaucratic log-jams, judges may not be happy about hearing these types of cases, etc.), it is doubtful this could ultimately prevent a same-sex couple from getting divorced in Texas.  Interestingly, the Texas Constitution  was amended to define marriage as technically between a man and a woman, however, that amendment is also being challenged, and will likely be reversed in light of the U.S. Supreme Court's recent ruling.  So, yes, you can get divorced.

Is the procedure for divorce for a same-sex couple different in Texas? 

No, ultimately, the process for a divorce for a same-sex couple will be the same, however; the pleadings will not say, "Husband" or "Wife," but likely refer to the parties as "Respondent" or "Petitioner" throughout.  This includes the Original Petition for Divorce (the document that starts the lawsuit) and the Final Decree of Divorce (the court order signed by a judge that divorces the parties). 

Can I get married to my same-sex partner in Texas?

Again, the short answer is yes.  Now that the U.S. Supreme Court has issued it's ruling legalizing same-sex marriage nationwide, men can marry men, and women can marry women in Texas.  Government documents and forms will likely need to be updated to be gender-neutral, but aside from bureaucratic delays (which is hopefully temporary), same-sex marriage is legal in Texas.

What about custody?  Can I get custody of my children, or modify custody, if I am LGBT in Texas? 

Yes, assuming that you already have parental rights of your child.  In Texas, before you can modify custody, or apply for custodial rights (in Texas custody is referred to as conservatorship), you must have standing as a parent to assert those rights.  If you are LGBT, and were granted custody and/or visitation rights by a prior order, it is likely that the court can now modify that order here.  Technically, any person, a parent, grandparent, aunt, uncle, brother, sister, cousin, LGBT party, etc., must first have legal standing to assert a parental right before they can ask the court to modify or even grant those rights.  Obviously, a case involving an order, birth certificate, or even just physical custody of a child, giving an LGBT party parental rights will have specific issues that need to be addressed with care.  Of course, the normal legal standards for custody and visitation factors will have to be taken into consideration, but outside factors like the county in which you file in (conservative or liberal), the judges you may appear before, and even considerations about the nature of the other parent's views on LGBT people and lifestyles will have to be seriously examined and discussed with your lawyer.