Thank you to everyone who made such thoughtful suggestions ! I really appreciate it.
I ended up narrowing it down to a luxury (or almost-luxury?) resort in Woodinville, the Willows Lodge. The rooms with balcony and fireplace looked soooo appealing. Obviously that wwasn't any near a $200-per-night limit, but I had some minutes of thinking what the heck - plus it was not so badly positioned for my last night, when I want to be in Seattle to see my niece at UW.
Then I started looking around on AirBnB. The place I really wanted to go was up on Guemes Island, but that seemed awfully far to go, and anyway I have been to Anacortes and the San Juan Islands before (not that Guemes is in the San Juans).
Then I was basically down to places on Vashon Island and near Gig Harbor, I figured getting to either of those would be relatively easy from SeaTac, and then when I’m done with my more “rural” part of “vacation,” I can just return the car to SeaTac and catch the light rail up to UW for the last night and back to the airport the next day.
In the end, I have opted for (reserved and paid yesterday) for an AirBnB on Fox Island. It is an extra small apartment on the land with someone's house. Very private, but with apparently lovely hosts who have thought of everything, a view of the water, and a lovely pine forest off the back/bedroom deck (lovely for morning tea). So I will go there the afternoon I arrive, and spend three nights there (so only two full days). That will be some nice time on the island and going into Gig Harbor, I imagine.
Then on Saturday, I will drive back to SeaTac and drop the car, and head into town, where I’ve reserved a Hilton on points. Will see my niece Saturday afternoon/ evening, then head out to the airport Sunday morning for that direct Seattle - Tulsa flight ! (They started running this route back in March, which is what gave me the idea for this little stop.)
It's not much, but I’m looking forward to it. Hoping for those gorgeous October days. But even if I run into dreary wind and rain, I will be snug and cosy up in the little apartment.
Thank you all again - I wish I could do everything that you all have noted — there are so many interesting places to visit out there !!
And p.s. CL: good reminder to pack my good raincoat !!