I think all of the Seattle Public Schools that have real tracks have FieldTurf soccer fields on the inside and are scheduled heavily with school and club soccer. I could be wrong though.
SPU is a good suggestion. Other than their track practices, there is not usually much else going on except intramural softball and maybe some other intramural sports. If you ever trekked out there and the track was unavailable, you could just walk up to Queen Anne Bowl and get some dirt track action.
I can't believe I forgot about West Seattle Stadium. Yeah, it's packed with schools this time of year, but later in the evening it ought to be less crowded. It is owned by the Seattle Parks Department (though they are trying to get rid of it) not the Seattle Public Schools, so it is officially open for public use anytime it is not reserved (and not in the middle of the night obviously). Even if there were team there later in the evening, I would think you would be OK as long as you stayed out of their way. The parking sucks though.
I was coaching at King's in Shoreline up until December when I got married and had to move away, and I can tell you that they have started locking the track whenever it is not being used, so that would be tough. One of our coaches is usually there on the weekends around 3pm or so, if that is your neck of the woods.
Ballard HS just finished their track, same issues as all the other schools, but that is another rubber track to keep in mind.
If you head over to UW, you might try Dempsey Center in the evenings too. As the varsity teams start heading outside, the facility starts to open up. If you are just running around the track and there aren't any varsity teams in there, they'll usually leave you alone unless the person is coming who locks things up at night.
Here is another good suggestion: Lakeside High School. They are just off I-5 and 145th. They have a 6 lane rubber track. There is no fence around it, so it should be really easy to access in the evenings, and you can park on the street right in front of it (or they have a parking lot). They have a really sweet mix of pullup bars, parallel bars, etc over by the pole vault pit too.