Hey @jbomforman! As @REI-JenK mentioned there are some truly remarkable places to backpack in WA state. The answer to your question will vary on your level of experience and interest in some colder season hiking. There will undoubtedly be precipitation - be it snow at higher elevations or rain in most parts of the state.
This article by the Washington Trails Association: 13 Winter Backpacking Destinations has some really good suggestions. If you want to stick to the lower elevations to (potentially) avoid snow - check out some of the classics on the Olympic Peninsula, such as Shi Shi Beach and Cape Alava Loop (AKA the Ozette Triangle).
Having quality rain gear and warm layers is a must! We have great Virtual Outfitters if you need some tips and suggestions for layering systems and beyond. Glad to answer more questions that you might have, and happy that you're looking to explore some of the gems of the PNW!