If you fly into Seattle, you can take a couple of ferries across the Puget Sound and see some of the Sound that way
http://www.wsdot.wa.gov/ferries/info_desk/route-maps/
or you can drive down around the Sound, and out onto the Peninsula.
Winter can be be very unpredictable weather. You might luck out and have relatively great weather or great stormy weather. Many less expensive (and of course more expensive) lodging options are there. The winter offseason rates are good for beach houses. You can walk the beach in a storm, then go get into a hot tub, cooking your meals in the kitchen, or having meals out (though there aren't a ton of places around).
Summers are naturally more crowded.
A trip on Hwy 101 around the Olympia Peninsula is great. About 3-4 hours without stops. The northeast side of the peninsula, in the rain shadow, actually gets an average of about 10" of precip/ year in Sequim, WA (pronounced "Skwim"). Seattle gets around 30", Olympia about 45", Aberdeen, over 60".
You can Ferry it up to Vancouver Island, BC as well.
The Quinault Reservation has some great hikes and tide pools, plus very few people, as many people don't know that you just go to the Tribal Police station for a cheap permit to access Tribal land.
This is called Tunnel in Rock, or something like that. The open ocean is seen through the tunnel. A narrow 9 mile 7" clearance track is drivable, until you get the the place where an Astro Van slide off the road where the road collapsed. From there a mile hike on a very not maintained trail leads to the beach. I am not sure if Sasquatch sabotaged the Astro or not.
Speaking of which
http://www.northamericanbigfoot.com/
I am acquainted with this guy, co-worker of friend's boyfriend. A teacher by day at a charter school. He filmed a pilot for some channel like Discovery or TLC or something. I don't know what's happening with that.
We were waiting for road construction on the Peninsula, and I got out of the car to talk to the construction workers while waiting. I asked about Sasquatch sightings. The white guy just laugh a second. The Native guy, without blinking, said how his sister has seen them, and you have a much better chance of seeing them if you are downwind. You can smell 'em (stinky) and they don't smell you.
We had a mountain lion both on our peninsula, which is semi-rural, and in the parking lot of the west side Olympia Safeway. Grizzlies in NE WA. Its possible that there's bigfoot out on the peninsula or BC or OR.
http://www.youtube.com/watch?v=TaPjNQ1xeU8
We've stayed here twice, offseason rates were not bad. Big beach fires are great. Close to the Quinault Reservation beaches and trails, and not too far to Lake Quinault, which has some record holding type trees, or at least close enough.
Camping options are there, too, but only summer would be fun.