I went there last year on a Wednesday before the 4th of July weekend. While it is true it gets packed quick, both Wednesday and Thursday there were still spots available. I think you'll be fine if you get there early enough (the sweet spot of the weekend campers leaving). Good luck!
It is always packed it's pretty much first come first serve you have to show up pretty early in the morning either on a Thursday or Friday to get a good spot it does get filled up quite fast unfortunately
Buckhorn is close for camping, usually opens around Memorial Day weekend.
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