Laundry in Skagway, located at Second and State does have drop off service, but I have never used it and always use the self serve. I have had some splended conversations with people.
Anyway if you use the big propane dryers you can be done in a little over an hour. You could contact the visitor site and get the laundry contact and inquire of rates. www.skagway.org
As for sailing in August- no it's not too late, I never book more than 60 days ahead myself. But if you are set on a cruiseline or ship, you need to inquire and book soon. It can pay to be flexable.