Though I don't have a gazllion x-country rides to my name, when I find that I do need to get off the bike and get some rest, I don't have any issue with pulling into a rest area and sleeping on the ground...something about having been a paratrooper in the infantry for 9 years...
When you do take a quickie, either keep it to about 45 minutes or at least 2 hours, since at about the 1-hour mark, you will hit REM sleep and if you don't give yourself enough time while in REM, you'll feel worse than before.
As for showers, well, most truck stops have those available, as well as having a dark room where you can grab a chair and sleep.