I think a weaker Civil Rights Act passes, no stronger than the one introduced in 1963, which means the EEOC is only an investigatory body with no employment discrimination provisions as in OTL's Title VII. As Kennedy was considering, he dumps Johnson and replaces him with Gov. Sanford of North Carolina. Needless to say LBJ is P.O.ed. and doesn't work very hard for the Democrats that fall. Not only does Texas go Republican in the Presidential Race, but George H. W. Bush wins his Senate race against Ralph Yarborough, However Kennedy/Sanford still have a landslide against Goldwater/Miller, it's just not as strong.
Kennedy (D-Mass.) / Sanford (D-N.C.) 421
Goldwater (R-Ariz.) / Miller (D-N.Y) 117
Besides, Texas, other Republican gains/holds in the Senate that didn't happen in real life would include Nevada, New Mexico, New York, Ohio, Oklahoma, and the special election in Tennessee, tho Gore Sr. would be reelected in the regular Tennessee election. So instead of a two seat gain, the Democrats would lose four to have only 62 Senate seats, five short of being able to override a Republican filibuster. (The Senate still had the two-thirds rule in those days.)
In the House, the Republicans gain or hold the following 31 seats, leaving the Dems with only a 264-171 majority:
Alaska - At Large
Arizona - 3rd
California - 22nd
Colorado - 2nd, 3rd
Connecticut - 4th
Idaho - 1st
Indiana - 11th
Iowa - 1st
Michigan - 2nd
Minnesota - 6th
Nebraska - 1st
New Hampshire - 1st
New Jersey - 2nd, 3rd, 9th, 12th
New Mexico - one of the two At-Large seats
New York - 3rd, 27th, 34th
Ohio - 1st, 3rd, At-Large
Pennsylvania - 19th
Tennessee - 9th
Texas - 5th
Virginia - 3rd
Wisconsin - 1st, 6th
Wyoming - At Large
North Carolina's 4th would almost have been a Republican gain, but the presence of Sanford on the ticket helps the Dems enough in North Carolina for Cooley to retain his seat and the Chairmanship of the Agriculture Committee.
The 1968 and later elections depend hugely upon how Kennedy approaches Vietnam. It's doubtful Kennedy would have gotten us as involved in Vietnam as Johnson did, but that still leaves a lot of wiggle room.