Presuming it gets abolished by a President Humphrey with a POD in 1966 or so, I predict 1976 is the first re-aligning position:
OTL: 321 - 217
1969-1973: Vice President Hubert Humphrey(D-MN)/Sen. Ed Muskie(D-ME)
1973-1977: Gov. Ronald Reagan(R-CA)/Sen. Bob Dole(R-KS)
In 1976, the tickets are:
Minority Leader Gerald Ford(R-MI)/Sen. Mark Hatfield(R-OR): 51.9%
Rep. Morris Udall(D-AZ)/Gov. Jimmy Carter(D-GA): 47.6%
I predict that the Presidencies goes something like this:
1977-1985: Gerald Ford(R-MI)/Mark Hatfield(R-OR)
1985-1993: Sen. James Carter(D-GA)/Sen. Bill Bradley(D-NJ)
1993-2001: Sen. Paul Tsongas(R-MA)/Rep. John Kasich(R-OH)
2001-2005: Gov. Hubert "Skip" Humphrey(D-MN)/Sen. Jack Carter(D-NV)
2005-2013: Gov. Howard Dean(R-VT)/Sen. Condoleeza Rice(R-CA)
2013-PRES: Sen. Jason Carter(D-GA)/Rep. Kurt Schrader(D-OR)