The governments of the the fifty states pretty closely mirror that of the national government, containing a legislature, judiciary and popularly elected executive. The largest difference appears to be that certain states eventually adopted a unicarmal legislature, although certainly not all (or even a large number).
It's interesting to note that the constitution does not dictate that this is so, and, to my knowledge, places no restriction on the type of governments of the state (save that they obey the rights to citizens given in the constitution itself).
You challenge, with a POD no earlier than the ratification of the constitution, is to produce a TL where there is the greatest diversity in state governments possible. I want some state to try to recreate a Roman consular system, one to go full on parliamentarian with a speaker/ pm having the most power. Have fun, and just think of all the different variety of republican institutions that could exist
One thought; since the basic American framework seems to be inherited from the colonial legislatures and governors, perhaps the best places for people to experiment would be in the Old Northwest and and going out from there.
Added points for any proposed, but abandoned, forks of state government from OTL to be implemented in the ATL.