I also bought my Catalina 27 in Port Orchard. Small world
Fortunately I only had to deliver my boat to the Vancouver area.
You are lucky in that your destination (I'm assuming it's where you live) is comparatively close. You should check into the coast of just trucking it, because then you can just get it done and you have your boat right away.
If you sail, expect to wait for weather windows to line up with your schedule. What is your schedule like? Will you be doing this on weekends? How flexible is your work?
Most likely there will be a couple of places where you will have to stop and wait for the weather. Ideally you would be able to get home from these places and return the following weekend to continue. While the WA coast is not exactly pock-marked with safe anchorages, your longest leg, from Neah Bay to Grays Harbor, is about 100 nm, so should not be more than two days, assuming you have wind the whole way and plenty of crew to keep the boat going 24 hours a day.
That's not to say there aren't other long stretches, but most should be doable between sunup and sundown if you're going in June, which, weather-wise, is probably a good bet.
Regarding going in May, just judging from the weather we get in our local protected waters, I'd say June is probably marginally better. I'm guessing you're not interested in paying for moorage in Port Orchard but have already paid for April. If that's the case, consider the difference in cost between another month of transient moorage and shipping the boat by truck on highway 5. Don't forget to add in the cost of transient moorage for a week at a time at all the places you have to stop.
Personally I think sailing the whole way would be fun, and certainly mostly safe if you play your cards right, but having done a long "commuter cruise" delivery before when I didn't have a full-time job, I'd be considering different options at this point.