Definitely do a search for long term parking. Not all lots/garages allow parking over a week.
I agree with parking at the airport. It's a little time consuming, but it will save you so much money! Take the lightrail back to WM. I think it's $3.75. 40 bucks per night vs about 10 for parking. If you are local, as in living in western Washington, take the bus to the airport! Seniors are $2 and adult is $3-$4
There are a number of parking garages in that area but if you are staying at the Camlin they do have a valet parking which I believe is about $40 a night, but the are other self park garages that are cheaper.
The best way would be to park in a secured lot by the airport and take an Uber or the light rail up to Seattle. Watch out for street parking. It may seem OK, but there is a 72 hour limit that is enforced in a lot of areas.
South east of hotel there is a multi story parking garage.
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