Mammoth campground in the park itself is one of our favorites. It's open year-round but no hookups. Madison is another very nice campground - no hookups.
Driving into the Tetons is a very long day. Are you retired? Plan to add more days to your trip and really see the parks. Stay in both of them.
If you want to stay in Cody, you might want to consider Buffalo Bill State Park. Here's a nice review of it. There are some sites with W/E. I wouldn't think you need sewer for 5 nights.
https://wheelingit.us/?s=cody%2C+wyoming