First, I agree that if Nixon loses in '68, after also losing in '60, and losing the '62 governor's race, he's finished. He's Thomas E. Dewey and Harold Stassen combined. The Republicans would never touch him again.
In OTL, Humphrey had many private doubts about the Vietnam War. As early as 1965, he wrote LBJ a memo suggesting we rethink our strategy. LBJ ignored him. Publicly, Humphrey felt he had to support the administration position. Privately, he had a lot of doubts.
As president, HHH would have ended the war much more quickly than Nixon did.
Tom B is right. With Nixon out of the picture, a mix of Republican governors and senators run in 1972, people like Chuck Percy, Howard Baker, maybe Reagan or Rockefeller. It's anybody's guess who wins. But Humphrey probably beats GOP Candidate X, and serves until 1977.
A Republican beats Muskie in 1976. Voters don't like having one party in power too long, and after 16 years of Democrats, they'd be ready for a change. But the Republican elected in '76 would have to deal with the inflation and the Iran hostage crisis, and that Republican would lose in 1980.