You have to be clever, passionate, and treat it like a secret side-job. But you can do it. My wife and I found a place at the end of February after about 3-4 weeks, and I feel like I snuck into the VIP section at the super bowl.
It took us about a month and we probably looked at about 20 houses/units - mostly around west seattle and shoreline with a few around sand point and beacon hill. We made 2 offers in 2 weeks and were beat out by ~$40k cash offers in the $360k range(!).
We found a town house in Georgetown listed at $360, offered $375k, escalated to $385k. Built in 2002, 3br/2ba + utility closet for wash/dryer, storage, redone kitchen, garage, small patio practically next door to Oxbow Park. Couldn't be happier.
The unit was being rented by day sleepers, so an appointment was needed during very particular hours. We got in on the day of the open house (before the open house) and brought our inspector with us. It passed our inspection so we waived that contingency (and no others waived), which is what won us the unit despite higher offers on the table.
It is a crazy market and it isn't being helped by the blood thirsty real estate agents - our agent's boss suggested we waive all contingencies when we were about to make an offer on the 2nd place we wanted - a 900sf 2br in white center listed at $359k that sold for $405k cash. There was no way in hell I was going to waive contingencies on my first purchase.
You have to kind of treat it like a second job, get ready to educate the fuck out of yourself and get a GREAT inspector. Don't get lofty ideas thinking "oh we can fix the foundation, its $100k below our budget, we can flip it in two years!". Don't do that unless you work in construction.
Our agent was a good friend of ours (AND extremely professional) and we were her only client at the time, so we had her full attention plus she has a great network we were able to rely on for inspections, and she was also highly dedicated to us as clients. Every little thing adds up.
You'll find something, be flexible.