Consider Everett or Bremerton. Light rail is getting built north toward Everett, and Bremerton is closer to many outdoor activities, but only a ferry ride away from downtown Seattle.

Bremerton and Everett both have affordable options below your cap.

Western Washington has some hidden financial advantages. Like Texas there is no income tax. And you'll likely save on utilities. Winters are mild and so are summers. Many homes don't even have air conditioning.