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MAP.
If it were me I would drive straight through the park to Cody. Take WY120 to Worland and spend the night and let the kids play in the hot springs pools. You will probably like a soak as well! We prefer Star Plunge. Be sure to find an RV park in Worland and make a reservation! There is a small but very interesting dinosaur museum to visit in Thermopolis too!
From Worland I would take US20 to Worland and then US16 over Powder River Pass to I80 at Buffalo. Powder River Pass is a serious pass with steep grades, lots of turns and spectacular scenery. Millions of people in vehicles very similar to yours make this drive every year....You can do it!
Because I don't know where you want to go in the Black Hills I chose Custer, SD for your destination.
Don't forget to wave HI to all my relatives as you pass Gillette!
I suggest you take US385 all the way from Custer, SD to Vega, TX where you can pickup I40 directly into Albuquerque. I would forget I25!