I'd highly suggest you skip staying in expensive Jackson and head into the park itself for the best overall experience. That's why you're going there anyway. There's Colter Bay RV park or Headwaters RV if you need full hookups. If you can manage dry camping for a couple days then there are other options. Our favorite is Gros Ventre campground. Be sure to check out the park's official 'nps' web site for lots of good information on things to do, where to stay, special alerts, etc. (The same for your Yellowstone stay)
https://www.nps.gov/grte/planyourvisit/camping.htm