Texas was overall Democrat run, but remember Eisenhower carried Texas twice. Dallas County turned Republican in the 1950s, supporting Eisenhower by a nearly 2 to 1 margin in 1956. Bruce Alger, the Dallas Congressman in the late 1950s and early 1960s, was a libertarian, the Ron Paul of that time. Moreover, there were several extremely rich Dallas businessmen, such as oilman H.L. Hunt, who were sympathetic toward right wing causes. The John Birch Society, the deplorables of that time, was very active in Dallas.
If the Kennedy assassination were a Soviet and domestic leftist plot, Dallas would be the ideal location, as the "climate of extremism" could be blamed for the assassination. However, the capture of Oswald, with his Soviet defector background, would deflate that narrative.
I don't think the Soviets had the ability to carry out such an assassination in this country, and if they did, they would have used a professional assassin, not Oswald.