Honestly I find the idea the USA will cease to exist ITTL unlikely.
The American identity was already pretty strong by 1812. Even ITTL New England broke away based on simmering resentment and the British terror/proBritish outlook. Keep in mind New York ITTL stayed loyal to the Union during 1812 and it took a break along mostly the Free State-Slave State Divide to cause the next war. While I could see an FSA resurgence, I really don't see why the South would break up, or that the former FSA would not be a united polity claiming an 'American' legitimacy.
As for neighbors carving them up, that's rather limited. New England at most would take more former FSA territory, and even New York only joined them because it was seen as the lesser evil. And New England is already investing in colonialism so their time and money is not going to solely in the region. Mishigama reclaiming further lands in the Midewest is less likely every year as that territory becomes more settled and populous. Canada/Britain at most might take the new Northwest, and frankly with Britain's Asian and Pacific ventures kicking off there's going to be less interest in taking disputed territory in North America. Louisiana will not be interested in taking a bunch of Anglophone land, save for their Northern border their focus will likely be on developing/securing their western territories. And Spain, well the Florida expedition was likely their last gasp of expansion in North America with them having more promising prospects elsewhere.
As for the African-American population, sadly short of some border buffer state somewhere I don't see them being able to seize control, or anyone being very invested in letting them do so. The recent revolt shows how hard actually overthrowing the system would be.
So in short I see no reason outside the former FSA that Americans would see breaking up 'their' country in their interest. And I don't see the neighboring powers having the drive to carry out such a task. If anything I expect the South will be more united than ever after this war with State identity taking a big hit with them having endured loss and victory in this recent war.
My guess is that Calhoun will die earlier than OTL with Jackson becoming Dictator, who will end up moderating some of Calhoun's policies. Jackson's reign marks the height of the Post Civil War order, perhaps even taking advantage of a general European war letting him retake West Florida or Northern Missouri or something. But even successes are in the end too expensive in the doing to be worth it and the planters continued stranglehold ensures any gains are not to the benefit of the common American despite the nationalist propaganda. All leading to a revolution Post Jackson that sees a New American Order come to power.