When LBJ dropped out in '68 it was only half of a change in his presidency. The other half was to move towards negotiating the end of the War.
What if LJB hadn't dropped out and instead fully embraced ending the war, praising McCarthy and appointing him as the lead negotiator in Paris. McCarthy, having achieved his goal in running, drops out of the race. Kennedy never enters. Humphrey doesn't enter either of course.
Then in June LBJ has a heart attack and drops out then. McCarthy has already become a spokesman for the new anti-war LBJ and has lots of prestige, in both the young student left side of the party and the loyal to the President side. Kennedy has no ground game and so doesn't enter, especially when McCarthy's people let his people know he has the VP if he wants it or Secretary of State.
Humphrey is ready to jump in when LBJ, weak physically but strong in personality, calls him in, reads him the riot act, explains that a contested convention this late in the game will give Nixon the election and explain if he enters he will have LBJ have to endorse McCarthy over him, which will be bad for him. Humphrey does as LBJ says.
There is no assassination of RFK. There are no riots in Chicago. The Convention is a love-fest. RFK is VP. Humphrey is told that a seat on SCOTUS is likely. He campaigns vigorously for McCarthy.
Nixon's "secret plan" is exposed as the treason it really was when an LBJ more committed to ending the war in TTL than in OTL releases the information on Nixon's shenanigans with Thieu on delaying the peace talks.
McCarthy wins the election. In December a peace treaty is signed in Paris.
In 1968 and 1969 the deep south going to Wallace is seen as keeping the Democratic win from being a landslide. By 1977 and two more elections in which it becomes clear the deep south had been lost to the Democrats after 1964, the common wisdom becomes that Wallace taking the deep south kept Nixon from winning and is why McCarthy won.
Nixon is arrested in December of 1968 by LBJ's Justice Department. His trial is all the news in 1969. He is found guilty of treason and is sentenced to life in prison. He is especially despised by Vietnam Vets who believe if he hadn't been exposed that the war would have continued for years and they might have died.
LBJ had attempted to nominate Abe Fortas for Chief Justice of the SCOTUS in 1986 when Warren announced his retirement, but had been stopped by a filibuster. In 1969 McCarthy nominated Humphrey who was easily confirmed and held the office for 9 years until his death in 1978.
Butterflies:
1) Since there's no enemies list there's no attempt to kick John Lennon out of the country. His life is altered enough that he isn't assassinated in 1980. In 1985 the Beatles reunited to perform one short set during Live Aid. They only performed four songs, all solo material from their post-Beatles career: Imagine, Maybe I'm Amazed, It Don't Come Easy and My Sweet Lord. In 1995 when George needs money and the Anthology is made, all four Beatles cooperate to release new material.
2) Since the war was over there was no invasion of Cambodia, no student response, no shootings at Kent State. There also had been no riots in Chicago in 1968 and so no trial of the Chicago Seven. The Hippy counter culture and its radical activist sibling didn't turn inward and didn't retreat to communes and food co-ops. On the other hand a small segment of it didn't turn radically violent either. Instead Flower Power continued to be a part of the American cultural scene with it's main political focus being anti-nuclear weapons and environmentalism.
3) With his brother alive and a powerful Vice President, Teddy Kennedy chose to retreat from public life after the Chappaquiddick accident instead of his choice in OTL to remain due to being the heir of the legacy.
4) There is no drug war. Nixon's drug law that federalized all drugs in certain schedules, which were decided by administration instead of legislation, never happened. Marijuana legalization was entirely a state issue and by 1979 five states had fully legalized it. By 1986 thirty-three states had. In 2012 it only remains illegal in Mississippi, Utah and Kansas.