Who says Lincoln can't run for a third term? He would be 59 in 1868. I would go for Lincoln/Grant.
Lincoln and Mary's plan was to go out to California and explore the west. Cute really. Just an old couple lounging around the western country as they travel to a fro, seeing San Francisco and the Pacific Northwest, photo-ops with Mark Twain and Abe living out the rest of his days quietly and peacefully compared to two terms which aged him decades by deep melancholy. Sad how it didn't happen. So term 3 is a no go unless Abe somehow feels the nation really desperately needs him.
Reconstruction may be somewhat easier on the south with less of a revenge factor, but I can still see Abe sending in troops to occupy the south to make sure it recovers and puts in place civil liberties for the freed Negroes, at least after they prove unwilling to do so themselves (which is likely). I also think his plan was to educate the Negroes via freedmen schools and then give them citizenship and the vote so he may have educated and then given that, or given that while educating them, or maybe events may have pushed him to deal with it some other way such as how the "Radical Republicans" did it. Perhaps you'll see
Grant still; I'm not sure who any contenders could be since wikipedia says the GOP rallied around him unanimously.
Who runs in 1868, I don't know. But its not going to be Johnson for the GOP. Johnson was a Democrat and the Republicans only ran him on the temporary coalition party of the "Union Party". Post 1868, there will be no such coalition. I do think Lincoln could coax the GOP not to wave the bloody shirt against the Democrats, though. Or at least he'd try.
*****
Anywho, how would this affect political realignment? Would the Republican party remain the progressive most party and the Democrats the conservative most of the two if Lincoln lives? Or would history find a way and the Republicans would still go into the conservative camp and the Democrats the progressive/liberal one (that also leaves the issue of "factions" (liberal, moderate and conservative, which had existed in both of the two parties and only faded in the last few decades of our history; I think Reagan hit that death nail.) and whether they'd still exist or fade away as they did in the OTL leaving only a conservative party (Republicans) or liberal one (Democrats), but that may be too complex to gauge.)?