I can see the case for the CSA keeping slavery into the 20th Century, but not the South within the US. It's such a horrifying institution that the US simply won't accept being a pariah state. They'll manage to get it banned somehow, be it via an activist Supreme Court decision, splitting the West up into numerous more states to get a larger free state majority, a popular campaign to flood marginal states with abolitionist settlers, or (probably the most likely) huge earmarks to a few slave states as a bribe to get them to side with them.