We are looking to make a 'practice' run this Spring, making a cross country trip for 6-8 weeks. It could very easily lead to a full-time period, depending on how it (and some work items) works out. ...
Have you checked shipping rates lately? I don't think you could ship dirty clothes home/clean clothes out cheaper than you could go to a laundromat. You might try laundromats in smaller towns rather than large towns you might pass through. Some campgrounds have cheap laundry, some are more expensive. Either way, it's less of an investment in time. You can do 4 loads as quickly as one load. An hour and a half later, you are good to go for another 10 days.