Wrong
Texas was blue in 1960 and stayed that way until the late 80s and early 90s.
Yes, there were a lot of Democrats in Texas in those days—when LBJ won the Senatorial primary in 1948 (by 87 votes) it was tantamount to winning the election. But John Tower won the seat that LBJ vacated when he became Vice President and the Kennedy-Johnson margin in 1960 was just 46,000 votes.