When you are traveling cross-country in good weather, you can usually find places to camp or sleep in your vehicle. The main drawback to this kind of accommodation is the lack of shower facilities. Where do you find a shower without paying the full price of a hotel room. Truck stops!
Most long-haul truckers sleep in their vehicles. That's the function of the big box behind the cab: It's camper! There is rarely a shower in there however, so truckers check in at their friendly truck stop for a showers. Showers at truck stops are sometimes free for truckers who fuel up at that location, but ordinary motorists can usually use the showers, too, for a fee. The cost ranges from $5 to $15 (which is at least better than getting a motel room at $40+).
A typical shower is just a private restroom with a shower stall. Towels and soap are usually provided, and you can usually take as long as you want inside.
A lot of rural interstate truck stops have them -- nearly any place with a lot of trucks in it's parking lot. The showers may not be advertised, but if there is a trucker's lounge, there is a good chance there are showers.
The major truck stop chains specify on their websites whether showers are available at each location. Try Pilot Travel Centers, Flying J, Petro Truck Stops and TA Travel Centers.