If all you're trying to accomplish is to get to Wyoming, not see anything enroute, and not need the TM in Wyoming, then I think I would lean towards hotel if you are doing the trip by yourself, especially considering you are new to campers. I just think the hotel route would be less risky, less frustrating, and would also likely get you there faster since your travel days could be longer and your speed higher. But, it wouldn't be impossible to do in the TM, and as long as you gave yourself plenty of time and had the expectation for some hiccups along the way, you'd be fine.
However, I bet you could find hotels along the way that would take 2 dogs if you did some advance planning, and like you said, not only are gas prices high these days, but so too are fees for campgrounds that are close to interstates -- minimum of $50, and could be as high as $80. Of course, if you're at a public campground (county, state, national), it could easily be far less expensive, around $20 or even free, but those places are not typically close to interstates.
Hotel rooms, meanwhile, along interstates, particularly in relatively sparely populated areas between Indiana and Wyoming, are pretty cheap. Maybe $100? Wouldn't take too much gas to make up the $50 difference just between a hotel and campground. I've traveled cross country in a car (no TM) before with a one dog multiple times and stopping at hotels, and it was easy to do, though that was 20+ years ago.
Dave
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