We took a trip to our favorite area last Fall, the Grand Teton and Yellowstone area. One campground had more campers than normal, but we were still able to get in and juggled another night to move to a better site. At another CG near West Yellowstone,

Our camp at the Tetons on the 4th of July weekend. Cost--nothing
As for Campgrounds inside the two parks: about 2 in 3 were closed, so yeah that made it harder to get a spot in the pasrk.
As for YNP. We stayed just north of Gardiner, about 10 minutes away. Cost again -- not a penny right on the river too.