Could the Cathars gone the way of the Eastern Orthadox and split from the Roman Catholic church and set up their own religion and states?
Well, Catharism did cut ties with Rome, almost by definition. The problem being that while Orthodox and Catholic churchs split over a very slow differentiation based on cultural and political differences and then theological; Catharism was a much more radical rupture without much in the way of big-scale support.
Medieval Langueoc was indeed a political puzzle, on which big names (such as Count of Toulouse) had an hard time being obeyed from their vassals (wheter lords or consulate) due to various crisis since the Xth century (two important ones being the War of Succession of Auvergne, and the Great Southern War). Catharism then developed on a hugely divided political landscape where notion of lineages and horizontal solidarities played a major role.
It's not that Catharism was ever majoritary
even in its main region : it was mostly an urban and peripheral nobiliar belief (and even there, it's generally assumed we're talking 10% of towns populations in Upper Languedoc, with cities as Carcassonne barely harbouring anhandful of them). But in such a political situation, social solidarities played fully,
while other dualistic takes on Christianity in Europe such as Tisserands found little social ground to blossom (at the exception of North Italy, where religious struggles allowed dualist groups to persist, but Rome's proximity was too huge of an obstacle)
So, before having "Cathar states", you'd first need to adress the problem of unifying a lot more southern polities. Which likely would have the result if done before to butterfly away the persistence of dualism in Languedoc. You'd then need to have a later PoD : we could imagine a more successful Raimondin participation (there's a TL about a successful Siege of Castelnaudary leading to reaffirmation of Toulouse's power over Lower Languedoc, for instance) but that definitely doesn't mean Cathars would be part of (in fact, a common grievance in the last part of the Crusade was that southern lords were on the bad recieving end of the Crusades because of Cathars who did nothing to help and were generally considered as nuisance. Note that, while not bent on repressing Catharism for the kick of it, as it was associated with refusal of feudal society basis, it wasn't that well seen by the greater nobles who went to fight with Crusader less to defend heresy, than to defend their nobility and preserve their own power.
While you could have a persistence of Catharism (which wasn't really unified theologically) on the same model than Vaudois, I really doubt they would be able to form their own states.