The election of Abraham Lincoln, a strong anti-slavery Republican candidate was the match that lit the powder keg of The Civil War. Elect someone else in 1860, especially a Democrat, you are going to delay The Civil War until perhaps the late 1860's or early to mid 1870's. Doubt you could delay it much beyond that.
But if someone other that Lincoln is elected in 1860, preferrably a Democrat like Stephen Douglas, I think you can delay that violent clash.
You can't get a Democrat elected in 1860 without massive changes well before the election. The Democrats splintered that year into the Northern Democrats, lead by Douglas, who were no longer willing to let the southern branch of the party dictate things, and the South Democrats who walked out rather than support Douglas and the Northern Democrats' platform.
After that, any Republican Party candidate would have won. And any Republican President would have resulted in the Southern leadership seeking secession. Changing the Republican nominee is not going to change that, and Seward, the most likely alternate choice, is even more unacceptable to the South than Lincoln.
But even if by some miracle the Democratic Party doesn't split, their man still doesn't win. If Lincoln had faced one opponent who got every vote that Bell, Brekinridge, and Douglas; he would still win the majority of the electoral vote and thus the election. In OTL, Lincoln got 180 electoral votes, Breckinridge (Southern Democrat) 72, Bell (Constitutional Union) got 39, and Douglas (Northern Democrat) only 12. Combining the votes of the last three means Lincoln loses California and Oregon, which swings 7 electoral Votes. Which means Lincoln beats Belkinridgelas 173-130.
Summarizing - the Republican candidate wins. Even if this isn't Lincoln, the south secedes, with little chance of success.