I think if you contact Swedish West Seattle on California as a new local with a kid with eye infection they ought get you in right away. They also have an MD on call at all times – if urgent, call now.
320-5780
The office keeps many daytime spaces open for urgent care/sick kid visits. You could talk about a primary MD for kid when there. Also, the MDs upstairs are fine as GP physicians. I’d easily recommend Dr. Dagg. 3rd floor same building.