Jimmy Carter in 1976.
It might be noted that since 2004, Democratic candidates have received more than 40% of the presidential popular vote in Texas. In 2008, Barack Obama received 44% of the Texas popular vote, and 41% in 2012. In 2016, Hillary Clinton received 42% of the Texas popular vote.
The last person to win the governorship of Texas as a Democrat was Ann Richards, who served as governor from 1991 to 1995. Republicans have dominated that office since then, but from the 1870s to the 1970s, all of Texas' elected governors were Democrats.