Tricky, but not impossible, lets not run Tricky Dick in 1968.
1968: Hubert Humphrey's not RFK is shot and killed, Kennedy really is innocent in the sudden death of Humphrey's. The Vice President was seen as too closely connected to the policies of the Johnson administration, and was shot by a Vietnam veteran who had returned home a broken man.
RFK is nominated the Democratic candidate and goes on to beat George Romney in November (A Catholic-vs-A Mormon). The United States gets out of Vietnam in 1970, more than less the same way it did in OTL, with South Vietnam being swallowed up by North Vietnam in '73.
RFK is re-elected in '72 defeating John Rockefeller.
In 1976 the Democrats put up Vice President Terry Sanford while the Republicans run Ronald Reagan. Reagan wins ending the Democrats long hold on the White House. Unfortunately for Reagan many of the same problems that plagued Carter still plague him, and when the Iranian Hostage Crisis occurs, he leads the United States to war.
Still reeling from the bitter memories of Vietnam, the US is uncomfortable with the occupation of Iran. Reagan is renominated in 1980 but is defeated by Ted Kennedy in November. Ted Kennedy only serves three months as President before being shot by Mark David Chapman in Los Angeles (rather than killing John Lennon to get famous he shoots the President instead). Vice President Jimmy Carter is sworn into office and proves ineffective in dealing with the occupation of Iran, which ultimately mars his time in office.
in 1984 President Carter loses in a landslide to Bob Dole in one of the most boring campaigns in history. A moderate Republican, President Dole is careful not to rock the boat and wins a second term defeating Senator Al Gore in THE most boring campaign in history (America found its cure for insomnia).
Vice President George H.W. Bush takes the Republican nomination but loses to Senator Paul Tsongas in 1992. Tsongas dies in office in spring of 1996, Vice President Clinton takes the helm and wins his parties nomination for the election. Clinton wins in 1996 against Jack Kemp and Lamar Alexander in 2000.
In 2004 Republican challenger John McCain is elected President in a more liberal America.