This coincides with the national shift each party went through beginning in the late 1940s and picking up in 1964 when the parties changed positions on civil rights. The South gradually switched allegiances from the Democratic to the Republican Party. The online Texas Politics text book has a sub-chapter briefly detailing the shift and offers this graphic:
Some observers argue that this reveals three distinct eras in the Texas party system. The first era dates back to the ratification of the 1876 Constitution and extends to the mid to late 1970s. This was the era of one party rule in the state. Republicans had little ability to win general elections, although the election of Senator John Tower to LBJ's vacant Senate seat following the 1960 presidential election foreshadowed a change. Democratic dominance during this era was absolute. There were some sessions of the legislature where no Republicans held seats. Any ideological divisions that existed in the state were contained within the Democratic Party - which had a conservative and liberal wing. The conservative wing did occasionally support the national Republican Party, so tension did exist.
From 1978 or so until 1994, the parties were in rough parity and Texas had something like a two party system in place. Democrats still held on to the legislature and competed with the Republican Party for governorships, but the shift favored Republicans and by 1994 - in-keeping with the national electoral wave that favored the Republican Party - the party dominated statewide offices, meaning the executive and top judicial positions. The Democratic Party is still competitive in legislative races, so they are not as weak as the Republican were prior to 1964, but Texas' is close to being the one party state it was at that time. Whether it remain so will depend on whether demographic trends that indicate a rise in the Latino population, which tends to vote Democrat, will bring the party back to power.
