Well, it is hard to say. Being the governor and everything, Shivers would have a good chance of unseating Senator Johnson, though the Democratic Party didn't really have any reason to unseat LBJ in 1954. Still, with Shivers being of the conservative faction which really dominated Texas, he did have a good chance at defeating LBJ. I'd be guessing he had a 50-50 chance of unseating Johnson.
If he did, he probably would have won; most of the South was reliably Democratic (in 1952, every state that Democrat Adlai Stevenson won was southern), especially for Congress, and it was like that until the early 1960s. In 1954, LBJ won re-election to the US Senate with 84%-14% against Republican Carlos Watson. If he unseated LBJ in the Democratic primary, Governor Shivers would have won. He had the experience, he was governor, and again, the state was reliably Democratic.
So, if Shivers is elected, he probably resigns the governorship sometime in December, maybe late November, and on January 3, 1955, is sworn in as U.S. Senator from Texas, succeeding Senator Lyndon Baines Johnson. With that, Senator Shivers could have led conservative Democrats to the G.O.P. pretty quickly and easily. I don't know how, but I'm sure that is possible.
So, if that happens, the Republican Party will have the majority in the Senate. What effect does that have on the Eisenhower presidency? That means President Eisenhower will have an easier presidency and an easier way of getting legislation past the Senate. Most likely, bills that were not passed under the Eisenhower Administration would be passed, and this could have a good or bad effect on the presidency, though, knowing that most of the bills proposed were OK, it would probably have a good effect on the presidency.
So, does this have an effect on 1956? Probably not much. Adlai Stevenson is probably nominated again to fight Eisenhower for the presidency, and loses again. There's no chance of him doing better than he did in '56, and two things happen in 1956: Either he does about the same, or he loses more states. Again, every state Stevenson won in 1956 was southern, so perhaps this "Republicanization," if I may coin that term, gives some more states to Eisenhower. I doubt that Eisenhower will win every state in 1956, but there is a really good chance that Eisenhower may win more states than he did, if that's even imaginable.
Then, what about 1960? Do to the massive defeat by Eisenhower in '56, Stevenson probably won't even run a third time, and if he does, he's probably ignored and doesn't win a single primary or delegate at the Democratic convention. Plus, Johnson isn't the Senate Majority Leader, and if it isn't William Knowland anymore, it's Everett Dirksen. If anything, JFK may have an easier time at winning the Democratic nomination for president. Most likely, Richard Nixon wins the GOP nomination, but who knows? Butterflies may see an upset for the GOP nomination by Senate Majority Leader Dirksen, though I doubt it. Most likely, Eisenhower has an even better approval rating, and considering it was already at a good level in OTL 1960 and Nixon lost barely in 1960 to JFK, he has a really good chance at winning the presidency in 1960. I'm guessing Nixon wins in 1960, and on January 20, 1961, is inaugurated as the 35th President of the United States and Eisenhower's successor.
Overall, it may have a big chance at changing history. So, that's my two cents on the subject.