In my experience, it was a lot harder to find free campsites in/near Yellowstone than it was in the Grand Tetons. On the bright side, with official campsites usually comes more amenities too.

I didn't stay here myself, but this campsite has 4.6 stars on google with 372 reviews, and the reviews say that there were clean bathrooms, showers, and laundry facilities, all of which might be helpful after all of that hiking you've been doing! On their website, it says this campsite is $45 a night.