I don't see him getting the presidency at all. The health/age issue would prevent it, plus the Dems might still be averse to a Southern nominee- even a Texan. Caro mentions how LBJ tried to rebrand as a Westerner in '60.
Texas is more Western than Southern, to be fair. If you were forced to be totally accurate, though, Texas is neither Western or Southern, and is uniquely just Texas. All that matters in politics is what the voters believe it is.
Agreed on LBJ's age in 1968 being too old to run. Whether 1964 would be a possibility, I'm not so sure, and I think that election would be totally dependent on Nixon's chances of losing in the scenario where Nixon wins 1960. He would be the incumbent but he would also be 3 terms of the Republican party in power, so there might be some chance of dislodging him.
On the subject, I think Johnson would live longer if he were never to have been president. The stresses of the entire world of blame being on him are undoubtedly a factor that harmed his health, and he fell into bad habits (drinking, smoking) immediately after he left office because of the stress and hardship he had gone through with the Vietnam situation. Add to that the normal quickened aging process and pressure on health that comes with being president.