JMHO, but if hunting and fishing is at the top of your list, you might want to consider one of the other Rocky Mountain states. Washington is a beautiful state with lots of diversity in country, fishing, and hunting opportunity; but the problem is it is the second smallest state west of the Mississippi with the second largest population. Granted the majority of the population is on the west side, but many come over to the east side (where I am - 2 hours south of Spokane) to recreate. Washington's elk hunting success is usually around 8%, deer around 25% (I work for WA Dept of fish and wildlife). I worked as a game warden in Wyoming (moved for a family reason) and had far better hunting in Wyoming (much higher success rates. When I was there in the '90's, elk about 40%, deer about 60%, antelope 95%), and still miss it. I think Wyoming, Montana, Idaho, Oregon, and Colorado (maybe others as well, these I am most familiar with; again MHO)would be better choices for the western experience and good hunting and fishing.
That being said, Spokane is nicer than many large cities, and there is a lot of public land north of Spokane with deer, bear, cougar, elk, etc. Northern Idaho is nearby, and Idaho has good for deer and elk, although elk success has dropped in some areas with wolf numbers increasing and timber stands growing older (less forage). If you like the Spokane area, I think the fellows that suggested living in Idaho and working in Spokane have the best idea.