'The park stay limit is 14 consecutive days. Reservations which exceed the 14 consecutive day limit may be cancelled and refunded.' From https://www.recreation.gov/camping/campgrounds/232479
Idk about Cosby campground but my grandfather stayed at Douglas campground from the time they opened until the time they closed about every year. Their are other in Jefferson county that will do monthly I'm sure just give them a call.
I think they will not do monthly. There is a limit you can stay I think maybe about 2 weeks and you have to leave and then you can come back. Not sure how long you have to stay gone before you can come back.
I don't know but they close in October open up in March
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