The Washington State Library is in charge of the Washington State Newspaper Microfilm Project and maintains on film a large collection of newspapers dating from the 1870s. A guide to this collection is Katryn S. Hamilton, Newspapers on Microfilm in the Washington State Library (Olympia: Washington State Library, 1980; not at Family History Library).
A guide for locating early newspapers is "Newspapers of Washington Territory to 1890." This was published in Volume 13 (1922), Volume 14 (1923), and Volume 26 (1935) of the Washington Historical Quarterly (FHL book 979 B2pn.)
The Family History Library has not acquired copies of newspapers for the state, but has a few published vital records abstracted from newspapers.
Washington Secretary of State website:'http://www.secstate.wa.gov/history/newspapers.aspx
Washington State Library: http://www.secstate.wa.gov/library/?