We've stayed in Cody, West Yellowstone (hotels) and at a B&B north of Gardiner. We hope to camp next time we visit. Yeah, hotels can be expensive and more so toward Jackson Hole. Have you checked out any of the KOA's to see if they have openings? They usually have camping cabins - just bring towels & sleeping bags. Buffalo Bill Village in Cody has cabins that are reasonable:
https://www.buffalobillvillage.com/
Definitely take your time in the park. Food inside the park is a little pricey & if you want to save a little, pack a cooler of food and have a picnic. Pack for all 4 seasons. I've been there in August when it's snowing/sleeting and another year it was 32 degrees as we drove in the park, though it warmed up. The last year we went, the weather was perfect. Get there early to see Old Faithful so you can get a good seat.
If you have the time, spend a day in Cody. Definitely visit the Buffalo Bill Center of the West. It is amazing and has grown over the years. I was there in 1981, 1998 and 2009 and each time it was different. Interestingly, they brought Buffalo Bill Cody's boyhood home from LeClaire, IA and it is now located at the center. Go see the western shootout outside the Irma hotel. It's kinda cheesy, but fun. You can reserve a seat, though it costs. Old Trail Town is pretty neat. They've moved historical buildings to the site, including a cabin used by Butch Cassidy, Sundance Kid and other outlaws. Jeremiah Johnson's gravesite is also there, along with some others. When we took our sons (they were 9 & 11), we went river rafting with Wyoming River trips and the boys talked about it for years. Buffalo Bill Reservoir/Dam was kinda of a neat quick thing to do.
Another thing I'd highly recommend is Beartooth Scenic Highway.
Not sure if you have boys or girls. When we got there, one of the first things our boys said about Yellowstone was "You can fart here and no one would know it!" LOL