Another option...
JFK is shot and wounded by Lee Harvey Oswald on 22 November.
Oswald lives, and in the course of the investigation, his roots within the hard left (including his time in Russia and attempt to live in Cuba) are revealed.
With JFK wounded, and the assassination clearly being aimed at JFK from the left, things start to get a bit interesting for liberals.
The Republican National Committe's chairman manages to get Rockefeller and Goldwater to agree to support Michigan governor George Romney, a pro civil-rights Republican but one who is seen as more conservative, as a compromise candidate.
Romney's Vice Presidential nominee is William A. Miller. At this point, the Democrats have a problem. JFK's wounds preclude him from running for re-electon. The Democrats then face a rough fight between Wallace, Johnson, and Humphrey. Johnson wins, but is so battered, and now knows that whoever he picks as VP will alienate part of his coalition.
When the votes are tallied, the Republicans have picked up 14 seats in the House, and three in the Senate. Among the winners of Senate seats are George Bush in Texas, Paul Laxalt in Nevada, Howard Baker and Dan Kuykendall in Tennessee, and Edwin Mechan in New Mexico.
The final tallies give the Republicans the White House, 190 seats in the House, and 37 seats in the Senate. Not quite a sweep, but it does set things up.
In Romney's first term, the Vietnam War starts to heat up, but when Romney tells the generals to "win this thing fast - failure and stalemate are not options," the Air Force launches Operation Rolling Thunder on 14 April, 1965 [in essence the OTL version of Linebackers I and II combined with the Cambodian bombings]. The Tet Offensive of 1966 is launched, but ends up as a military victory for the United States, at which point, the North Vietnamese government asks for a cease-fire. President Romney states the terms: The DMZ becomes the international boundary, and all NVA forces must leave Cambodia and Laos. In return, America would draw down military forces in Vietnam. It is presented as a take-it-or-leave-it offer, with the alternative being a resumption of Rolling Thunder.
On 29 April, 1966, the Paris Treaty is signed. The United States ultimately drew down until all that was left in Vietnam were advisors, the 9th Infantry Division, and a fighter wing of the United States Air Force (pretty much, another Korea situation).
Rationale
For the GOP to win in 1964, two things are crucial: JFK must live, but be unable to run in 1964, and the GOP needs to nominate ABGR - anyone but Goldwater or Rockefeller.
For the GOP, the natural candidate is George C. Romney. As a businessman, he would reassure conservatives, yet he was also pro-civil rights. As a Mormon, he would also benefit from not having any skeletons in his closet. Furthermore, Romney could turn any attacks on his religion against the Democrats.