Nathan - You're just fine in those highway rest areas. That's what they're there for - to avoid drivers falling asleep at the wheel. Yeah, as the other David says, most states prohibit tents. Once it's dark, no one is going to care if you roll out your sleeping bag under the most distant picnic table, but you might get kissed on the nose in the middle of the night by a dog let off-leash to do its business or a skunk looking for discarded food.
So I just sleep in the car. Officially, there's a 4 or 6 or 8 hour limit in each state, but that's just to give an officer leeway to move someone undesirable along. Don't set up camp, don't hang out a clothes line, don't litter and everyone's cool.
A few tricks: You want to park well away from the restroom building because it will get traffic all night. The truck parking area is noisier because they idle their diesels all night. I test for street lights in my eyes by reclining my seats and double-checking. Sometimes I'll move my car a few spaces to improve that.
Biggest tip (I've done this a lot): Don't try to save gas. Waking up cold and shivering (or hot and sweaty) at 1 am is no way to get a good night's sleep. A gallon of gas is $4 and that's a lot less than a $50-90 motel room. All night long in a 4-cylinder car is less than 2 gallons of gas.
One time I was driving I-70 near Saline, Kansas and I remembered that a particular rest stop had the double-whammy of hot water in the sinks AND the 10-amp, hot-air hand dryers (not just the 2-amp fast air ones), making it a great place to do a sponge bath and wash my hair. Then it dawned on me, "I've done this a lot, if I'm memorized rest areas 1500 miles from home).
KOA campsites will sell you a shower for $5 or so. If you put a water-bladder on the rear deck of a sedan, by day's end it will be toasty warm for a shower (when car-camping on a road trip, I travel with a bladder and shower-head adapter). But you can also walk into any McD's and fill up a bladder or water bottle with their warm water. Generally, south of I-80 McD's bathrooms are multi-stall so you need to be more uninhibited to get naked by the sink and sponge yourself clean. Whereas in the northern tier of states, there's a locking door on a one-person bathroom, making it less nerve-racking to wash up. Service station bathrooms aren't nearly as clean at McD's, but are typically single user. And, HEY!, you just saved >$50 on a hotel room, so spend $1 on a roll of paper towels and whip down all surfaces, leaving it cleaner and drier than you found it. Road-tripping LNT.
Depending on how fast I'm aiming to travel (PB: 2650 miles Seattle-Kenai, 51 hours, solo), I'll stay in a motel every 2nd or 3rd night. A full-on shower and a flat bad is a nice thing. If there you two of you, you MUST bring a big pillow like on your bed at home. Then you can leverage the driving time with the other person taking a nap. Good tunes on ear buds and dressing a little cooler can help the driver stay alert while chaffuering the napper.