I think your best chance can be in state government colleges. But still you should go for NIT counselling,  in spot rounds it may work for you where it depends upon your availability at the spot.

Well that completely depends on whether you have a category or not. If you belong to general then may be apply in CCMT and see where it leads you but I doubt any seat in NITs . So, I would suggest you go for private colleges. But if you have a category then there are chances in NITs too.And for IITs I don't feel there's any chance but you may give it a try especially new IITs. So GOODLUCK!!

Best of luck.