Who has 6-8 weeks worth of clothes, even in their house to make all that shipping practical? That is 42-56 pair of underwear and socks, alone, per person. I assume that you are joking. Who has time or money to buy complete new wardrobes on the road? That would cost more and be more cumbersome than the laundromat.
Get a Kindle or a Nook so you can read or play games, buy some quick drying fabric blends, combine laundry with other errands. It's just part of living, like grocery shopping.
Or, you could alternate between a cauldron and a washboard, and going to the river to beat your jeans on a rock.