I've done Yellowstone three times now in August, including the eclipse frenzy last year. The other two visits were on the spur of the moment.
Since you're not a slave to hookups, you shouldn't have much of a problem at all. Just find a one night reservation at any campground in the park, say Madison. Leave your campground at 6am and head to one of the first come, first served campgrounds like Norris. Find a site folks are packing up at, and bingo, you're set for the rest of your stay.
As to crowds, yes it's busier in August than June or September. Just get up early, see the sights, then settle in somewhere for the afternoon. Follow that advice, you'll be fine. I used to avoid August because of all the stories about crowds. Still prefer June/September, but given the choice of going to Yellowstone in August or not at all that year, I'd rather be in Yellowstone.