Out of curiosity, I went onto recreation.gov and started the process of reserving a campsite at Buck Hall. My pass would have gotten me 50% off the cg fee, so I would have paid only $10.
See screenshot below. Note that the "America the Beautiful Annual Pass" does not apply for a discount. My pass looks like the one second from left.
Again, you must be 62+, so that may not apply to you (yet)