Consider reservations at busy places but not for most locations. Otherwise, you get into a situation where you feel pressured to go on when you may be tired or weather is horrible.
Also, DC isn't really on the way from NYC to Seattle.
No worries finding a laundromat. Go to a big one and you can use 3-4 washers at the same time and it will take 1.5hrs and you are on your way.
As long as the linens and kitchen stuff is a modest amount, not a huge issue flying it back. You might consider supplementing with stuff from the goodwill and leave it behind.
