As strange as it may sound, LBJ keeping the US out of Vietnam might have cost him a second term.
Middle America only really turned on the War when the Tet Offensive began. Until that point, the only people to oppose it were general isolationists (who were small in number thanks to the whole Cold War thing) and pacifists (who only really gained support amongst the youth when the idea of being sent to die sank in), and in fact, the vast majority (65% by some estimates) supported the Gulf of Tonkin Resolution. As a result, we'd likely see a massive backlash against LBJ for daring to let Vietnam (and by extension all of south-east Asia) fall to Communism. This backlash alone., coupled with the lack of a window to OTL that would exonerate the man would cast suspicion that he was a closet Communist, something not helped by his "Great Society" policies being redistributive and government interventionist in nature.
This alone would probably grant whomever wins the '68 Republican primaries an OTL '84-esque landslide victory, coupled with massive Congressional majorites which proceed to drive headlong into whichever 3rd World Nation is currently in the process of being overrun by Reds at that point (my guess is Cambodia), and see it turn into a quagmire probably on a larger scale than OTL Vietnam ever was, while simultaneously scaling back all of Johnson's achievements (outside of the Civil and Voting Rights Acts) if not scrapping them outright.
That said, on the off-chance he wins a 2 1/2th term, I can see him embarking on an attempt to pass a standardised welfare framework (think something akin to a Universal Basic Income) and a system of Universal Health Care, coupled with Health and Safety legislation, and possibly something like the Equal Rights Amendment, depending on how many chambers of Congress are under Democratic control by 1970.