AUSTIN, Tex. — Fourteen states allowed same-sex marriage when Cleopatra De Leon and Nicole Dimetman filed a lawsuit late last year challenging Texas’ constitutional ban on marriages that are not between a man and a woman.

The couple, who had married in Massachusetts in 2009, live in Austin and collided with a Texas law when they could not both be listed on the birth certificate of their first child. They hope the law is changed before the birth of their second.

“There are moments in our lives that are flashbulb moments,” Ms. De Leon said during a town hall event last week, describing the decision she and Ms. Dimetman reached to challenge the state’s same-sex marriage ban.

With the case heading to the United States Court of Appeals for the Fifth Circuit in New Orleans next month, Ms. De Leon, Ms. Dimetman and gay rights advocates are hoping that Texas is on the verge of its own flashbulb moment.