In that era, the South, including Texas, was very supportive of the Democratic Party. The shift is relatively modern. Here's the House of Representatives only 25 years ago. Southern Democrats were conservative on social issues and a bit more conservative than other Dems on economic issues, but it wasn't like people imagine. They were plenty supportive of big government.

Texas used to be a democrat leaning state in presidential elections until 1952 to 1976 when it became a swing state and from 1980 onwards a state that has been easily won by republicans ever since.

Statewide offices and for federal offices with the exception of as i mentioned for presidential elections, Texas still elected democrats to state/federal positions until the 1990s when the presidential trend caught up to the statewide trend.

Just so you know, democrats were liberal since FDR but their 50 statewide parties were allowed to be flexible from the ideology of the national party. However once the rift over civil rights act and the new left/sexual revoluion and disputes over religion and guns. The southern wing of the party began to lose voters to the GOP and texas was not immune.

LBJ was a part of the texas democrat party when they were still conservative. Texas in fact has always been politically socially conservative but once the rift in social issues persisted as the national party began to move towards social liberalism, the voters shifted to the republican party. Id say that Texas isnt really a democrat or republican ,state but a conservative one. And as republicans became conservative, it wasn't their ideology that changed ,just their party ID.