Actually, they offered Houston the position of retaining his office as Governor under the confederate state upon his acknowledgement of that government. Houston knowingly and intentionally declined and exercised his absence from the swearing in procedures. As a result it was determined that the office had been vacated, whereupon the Lt. Gov. became Governor.
As for Houston, he returned home and took some time off while his son enlisted in the confederate army. He soon concluded that Texas would be best on its own outside of botht he union and confederacy, though came to support the confederacy in name. In 1863 or thereabouts, he began planning his political comeback and was widely expected to make a run for governor of confederate Texas and likely would have won the office, but a premature death deprived him of it. Andrew Jackson Hamilton, the Military Governor of Texas during the Civil War -- now there was another Texan to be proud of!
Too bad he never set foot in that office until June 22, 1865 after the war was over! [end sarcasm]
I was surprised to learn that Hamilton Pool, a sinkhole near Austin, is named after him.