Also in addition to Little America (only the Flagstaff and Little America, WY have tubs as far as I'm aware) The other 2 I know of is Travellers Oasis in Eden, ID near Twin Falls and Boise Stage Stop. You can't go wrong at Little America either location. The staff is super nice and the rooms are spotlessly clean.
Sapp Bros in Salt Lake City also has tubs. They're huge! I think it may only be 2 of the shower rooms that have the tub. Just request it when you pay for the shower. Best shower rooms I've had at a truck stop!