No to Yellowstone Valley RV for seeing Yellowstone Park. It's too far away.
Get reservations NOW for Fishing Bridge if you think you need full hookups. Otherwise, make reservations NOW for Bridge Bay campground which has water and dump station. You won't need a/c in the park. This campground is pretty well located for seeing Yellowstone.
Then make reservations NOW for Colter Bay campground in Grand Teton, either the RV camp with full hookups or the other one with water and dump station.
Now, for Cody, you can stay at the Ponderosa RV park or Absaroka Bay RV park. Or you can go 7 miles west and stay at Buffalo Bill State Park with elec, water and dump station..make reservations as early as you can.
Cody has a fine western museum worth your time and a gorgeous scenic drive you can do in a long day..the Chief Joseph drive and Beartooth Pass over to Red Lodge MT and then back to Cody from there.
If your heart is set on full hookups but you don't like Fishing Bridge prices, think about going over to West Yellowstone. There are several rv parks in town or close to it and you are right on the edge of the park.