Nice job copy and pasting the DOH website. :/
I'm currently, til tomorrow morning, a Washington state paramedic. I can give It to you straight. Here's the breakdown.
You cannot work in King County unless you attend Harborview's medic school. It's a closed program, meaning it's only open to new hires at King County Medic One or King County fire departments. You can't just apply and get in. On the west side, all of the 911 ALS, with the exception of King County Medic One, is fire based.
There are positions in Pierce, Kitsap, Skagit and Mason counties for some IFT and maybe some 911 medics, but they are few and far between.
Different story east of the mountains, but money for non fire medics sucks here. Think $14 an hour.
Unless you plan on testing for KCMO, and you want to attend Harborview's program and be a "mother may I" medic, don't plan on moving to Seattle to work as a medic.