Are you moving out of Seattle, or into Seattle instead. Either way, and wherever you are moving, you will need to get that address changed the right way to save you a lot of hassle later. Let the info and links below help.
Every post office has forms that will allow you to change your home address. This can be done at any post office, regardless of where your home is located. For a $1 fee, addresses can also be changed online.
After changing your address, the U.S. Postal Service will forward your mail for up to one year to your new address. Before that year is up, you should notify anyone sending you mail of your address change. You will need to contact multiple agencies to change your address with the government. Some of those agencies are:
USA.gov also provides a list of federal and state agencies to contact in other circumstances where you may need to change your address.