Your rental situation likely is governed by Chapter 59.18 RCW (Residential landlord-tenant act). The statutes are at http://apps.leg.wa.gov/rcw/default.aspx?Cite=59 .
You should also review your lease as many of its provisions carry over.
You likely have a month to month rental now since you have not signed a new agreement.
Unless your agreement allows you a longer notice period, the landlord's notice for you to end your tenancy likely is governed by RCW 59.18.200: "written notice of twenty days or more, preceding the end of any of the months or periods of tenancy, given by either party to the other".
If your lease has expired you are now on a month to month, and either party can end this tenancy with twenty days notice to the other party.
Twenty days does not adequately provide enough time to prep a move, and yet, that is what our Legislature has decided is adequate notice.
So the short answer is that your rights are somewhat limited. Your landlord has to give you 20 days notice on or before the 10th of the month. A 20-day that arrives later than that is untimely. For example if you pay rent on the first, then the notice is late after the tenth of the month.
Hope this helps - Elizabeth