A legitimate question about a singular Carolina...if it were to exist from the Constitutional era, how would that affect the paired admission of slave states and free states? If it is admitted early on, then the balance would be thrown off by the time we got to the 1790s / early 19th century. There would have inevitably been more free states overall.
Well, if we handwave how they unite and just make it fiat (the two are never split for some very odd reason, etc), then the presence of the Carolinas will, of course, upset any of the balance that comes along later, if the balance even arises. Once New York and New Jersey ban it (they were initially slave states, so you're going to have to count it in their column), we're going to have the balance swing to the Free States early.
The best immediate option might be to create another slave state. The best answer might be in Southern Illinois (Jefferson's famous Polypotamia) which shares some of those cultural relations to the south, akin to Missouri, while also being possibly open to the practice. That would restore the balance in the future while not expanding outside of the US's initial borders.
That might alter Illinois's northern border, though, and will likely cause further changes of development of the old Northwest, but it would be your best solution to restore the balance.
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One thing I was going to consider: say we have a Carolina Union and an indepdendent state of Georgia not confirm the new Constitution and stay independent for a couple of decades. While slavery would likely persist in the region, how would it be affected in the North? The only major slave state would be Virginia (later Kentucky) so the practice would be very difficult to continue.
I had imagined in my mind that the Carolina's/Georgia continue along indepdent for a few decades, but tied at the hip to the US. The US proper manages to secure Louisiana at roughly the same time frame, blocking off the two smaller states from Western expansion, while the US also eventually accepts West Florida into the Union as well. Follow up with a similar Adams-Otis Treaty,, and the US has completely encircled the states.
Slavery itself might wither on the vine in the US, maybe not being officially outlawed, but with nobody supporting it, it becomes de jure forbidden in most states while in many parts of the slave states it falls further and further out of favor. The protective tariffs favored by the North make it more unprofitable as well.
And, in the end, while the Carolinas and Georgia would make a majority of their money with trade overseas (esp Britain), the two countries fall on hard times in a recession, or a glut of the overseas market, etc. The citizens have always felt themselves to be Americans of a different stripe and many of the statesmen that opposed the Constitution then have fallen away, looking at a newer, pragmatic concern. (There's also a big issue in the west, as West Carolina/Yazoo, AtL Tennessee and North Mississippi/Alabama, might vote to leave on their own if the state as a whole doesn't join). So, for an assumption of their debts and a few other concerns, eventually Georgia and, in the end, Carolina join as federal units under some of the same guidelines as Texas would later on.
Just the thought I had in my head. Wanted to outline it.