That will depend on a large degree on your choices and the location of your tour. If you go somewhere with hiker-biker sites it should be pretty easy if you travel on the east coast it is just about impossible unless you stay with warmshowers.org hosts or something.

The hitch is the fact that you require showers every night. That may require either a room or an expensive campsite in many locations. On the Oregon and California coasts the hiker biker sites would make it possible. In the middle of the country staying in town parks might work, but counting on all of them having a shower is asking a lot.

Where are you planning top tour?

Maybe consider getting by with a wipe down with a wash cloth or save more money before your tour if you are not in an area where it might be difficult. Also if you are fussy enough to require a shower every night it makes me wonder if you might be fussy in other ways as well. I don't intend that to be an insult, but expecting a cheap place with a shower every night while camping is kind of unreasonable. In many locations it just isn't likely.