www.warmshowers.com
Cycle touring can be anything you want it to be. You can plan your tour around a motel every night and have your daily shower if you really want it.
When I'm stealth camping, I try to wash up at a public restroom sink before I stop. On a warm day, I'll even wash out my clothes and put them on wet. They dry quickly. I carry a small universal sink stopper since many public sinks lack them. On cold days, I'm usually not that sweaty so it's OK to skip a day or maybe four. Like everyone else says, keep the crotch clean and pay attention to your body.