You have lots of good advice. If you look at your map, and go across S. Dakota, then up the Yellowstone to Gardnier, you will have no big mountain passes to cross. I tow with a 6.7 Super Duty, like you, and the mountains are not a problem. I like staying in Gardnier at the campground right in town. The campground is on top of a big hill, but you can still walk to a garage and supermarket (at least in 2012).
Good fresh baked bread at the supermarket. Have the garage change the oil and maybe the fuel filters on a rainy or down day. They were reasonable in price in 2012. Bring your filters and synthetic oil for the Ford in case they do not have them.
If you are RC, go to mass at the log church in Teton Park. We got a Chicago Bishop out to Christen a family member. They were all real nice to a couple from lower Slobovia. Our jeans and travel shirts were a little out of place that day.
Add a week if you can, and do see the sights in S. Dakota. My NJ wife loves the Badlands, and I liked the girls with body paint in Sturgis (Motorcycle rally week). We saw Bighorn sheep in the Badlands, and big something else in Sturgis. And take the time to walk around Devils Tower. And drive the road less traveled.
Cheers