The fact that the Constitution flew through ratification so fast with half the states shows there was a strong political desire for some sort of remaining unity. So scenarios involving sudden, lasting, and even hostile divides among the 13 states are not terribly plausible, however entertaining they seem. The states that resisted ratification are your trouble spots--North Carolina, Rhode Island, Massachusetts, New York, and Virginia.
Potential for internal hostilities included Connecticut's land claims in northern Pennsylvania and its import policies undermining New York City's efforts to rebuild from the violent British occupation of the city, Rhode Island's reckless gamesmanship with Continental securities, which netted huge paper profits and screwed over outside speculators (rather than the acceptable tactic of simply screwing over the war veterans), and the armed debtors' rebellions in Massachusetts, Connecticut, and New Hampshire. The shipping relationship between Britain and the Southern states was also going to continue to create regional tensions that could lead to small regional unions or alliance. Interestingly, there was also great accord among the states over how to deal with the Ohio country--accepting that there'd be new states created there. Plus Spanish policy of locking down Georgia to within a fraction of its land claims was also a source of shared unification. A surprising number of early Georgia gentry were transplanted Yankees.
In short, there were as many factors pulling the union together as there were tearing it apart. Had the Philadelphia Constitution failed, there were bound to be other attempts at reconfiguring the nation. Philadelphia was, after all, the 3rd try in as many years. When you read Madison's Notes, you come away certain they didn't want to end up like the HRE or northern Italy, the play thing of the centralized monarchs of western Europe.