Do you have a species you are planning on focusing on? Within 4 hours of Seattle will get you into good Mule Deer habitat, and probably closer to what you are used to landscape wise.
There is decent Blacktail hunting and Roosevelt elk hunting on the west side, but lots of private timberland that requires passes. There is plenty of public land as well. In reality, most of the public land in Western WA has pretty good Blacktail numbers, it just takes getting in there and finding them.
The SW WA units like Williams Creek, have some of the best success rates on Blacktail, and you can find good bucks in there. This is one of the areas that is mainly private timber land though and you would need to purchase a pass for most of it. North and East of you is a fair amount of National Forest and DNR land. If it's actively logged, there will be deer around.
