This is a woman's project. I'm only giving a guiding hand in helping DS keeping mommy at least 20 minutes away. My brief stays with son is NOT looking for housing (futile for males, anyway) but seek some solitude from Mom and wife. Advised son to look for personal housing only when his "intended" has decided on the permanancy otherwise he's pretty happy where he's living-coop/shared living.
Off hand, I'd say areas south of Seattle, though there are probably some decent ones even south. If you drive around, it's pretty obvious where you wouldn't want to be. Stay away from the airport, Burien, maybe Des Moines, White Center. Most any other direction is probably okay. I'm surprised the you can't find something near the U if you're willing to go up to 300K.
DW thinking (most probably) moving to Seattle area when parents are gone.
Thinking of housing $200-300K condo with DS help and dependent on amount of downpayment. Like to stay within 30 minutes of Center of the Universe, by freeway, Link, or Bus. Area's to avoid? CotU and vicinity is too expensive for us.