I've bathed/showered for free at gyms, janitors closets at colleges, backyards with sprinklers left on during work hours, downtown fountains, hotel pool changing areas, churches, Walmart sinks, and if Im really desparate, ponds at city parks.
If you plan to be somewhere short term, you could always go on a YMCA or other gym "tour" as a prospective new member; they basically let you have free roam of the place for a day to "test it out" which also grants you access to showers.