So a Jeffersonian/Jacksonian South against a Hamiltonian North?
Yes, essentially that. The divide would naturally not be entirely geographic but given the higher support for Jeffersonianism in the South and Hamiltonianism in the North, I imagine that eventually a latitudinal divide would emerge, if nothing less than to get the bleeding to stop and peace to resume.
You could have a presidential election, where Jefferson and Hamilton are the candidates, with an 1876-style result that leads to civil war. The civil war ends in stalemate and so both north and south cannot conquer each other; eventually the war peters out Korean War-style.
Something along these lines was my main idea, specifically in regards to a civil war that peters out into a virtual stalemate with both factions emerging as independent republics.
That very much depends on what you mean by "north" and "south".
Where is the border? The Mason-Dixon Line? the Virginia-North Carolina border? The Potomac?
The parameters of what I had in mind here would mean that "the South" would more or less be joined by Democratic-Republicans in the north, but the strength of the Federalists there would cause at least the majority of the DemReps in the North to not carry the day, leaving a divide between the two republics along, ideally, the Mason-Dixon or Potomac. Though I did toy with Pennsylvania, at least the southern half of it from what was the old counties in the far west that Virginia claimed up to the Susquehanna River in the York-Cumberland County area, perhaps being in the Jeffersonian republic, haven't a clue if such a thing could work though and I'm fine with the Jeffersonian Republic just strictly being Virginia-Maryland and what's south and west of that.
Was there any discussion of a free womb law at the Constitutional Convention?
I'm not sure. I know Jefferson brought up the 1784 Ordinances to try and get all western states to be free states in an early display of free soilism. Though this is meant to be more a Jeffersonian Republic that is "Southern" insofar as that's where the DemReps could mount the strongest authority and rule rather than "Southern" in the CSA, slave-nationalist way. I'm explicitly trying to avoid too many proto-CSA parallels here, even though inevitably slavery will be the main industry of the republic.
The Northwest Territories will be of little interest to the South, but they'll want to hold on to their western claims.
Virginia's western claims included basically the entirety of the Northwest Territories which is why the question arose, and since Virginia is the most Jeffersonian of colonies at the time, it'd be a little weird to not have them in the Jeffersonian republic. My thoughts were maybe following the Mason-Dixon across and leaving the southern half to the South and northern half to the North or Britain, or creating some alternate line to effectuate a similar purpose if the M-D is a little too much of the Northwest Territories, mainly because the Ohio River was the main carrier of settlers to the Northwest at the time and most early settlers came by way of Virginia or what was later Kentucky but if that seems unlikely or too hard to achieve then I suppose the Ohio River being the border serves as a good natural barrier between the republics.
New Orleans won't be contiguous to either country because Florida extends all the way to Baton Rouge.
In regards to New Orleans, I was thinking a war between the Southern Republic and Spain would basically be inevitable, unless Britain takes N.O and then a war between the Southern Republic and Britain is on the table. The southwesterners will want N.O badly, the difficulty of getting commerce over the Appalachian Mountains is a very hard problem to surmount.