M.J.
Hi there! I'm not a single parent, but I was in the position of being new to thte area five years ago with a 3 year old and a new born... We're from Ohio so I know what you mean about better opportunities:)
Anyway, I would like to suggest that you look up the local Moms Club. I know there is one in Mill Creek because that was the one that I was in, but I'm sure that Lynnwood has thier own branch. It's agreat way to get started on meeting people and learning about age specific things to do around the area. they also set up play groups between moms with kids around the same age! if you google "Lynnwood moms club" I'm sure you'll find it.
Also there are all kinds of sites about places you can take kids around here that are free and fun! Do you know about forest park in Everett? There is a free petting zoo/farm there and a spray park and swim center.
Let me know if you'd like more info on anything:) and good luck!