As Caro's biographies point out, the RFK announcement was simply a convenient excuse for LBJ. He knew his health was bad and he knew he was a polarizing figure but, more importantly, he knew the Democrats didn't have a snowball's chance in hell in 1968 so he was happy to let RFK take the fall.
Johnson was perhaps one of hardest of hardball politicians ever to occupy the Presidency. He was also a brutally pragmatic in the political sense. He knew as president he'd burned through every bit of political capital he had getting the Civil Rights bills through, passing the legislative portions of the "Great Society", and expanding the Vietnam War. He knew the Democrats were doomed because a reaction to those policies was brewing and that, thanks to civil rights and his own experience with Thurmond candidacy in 1948, the South would go either with an independent or a Republican. In other words, he anticipated Nixon's "Southern Strategy" well before any other national politician was even aware of the possibility.
Leaving his health aside as he would have barely lived out his 1969-1973 term in office by all of two days, having Johnson run in '68 will require major changes to the political scene and not just to Johnson himself. You'll need to prune back civil rights, the Great Society, Vietnam, or more likely a mixture of the three in order to make Johnson less polarizing and retain some of his political capital.
Bill