Lots of hunting in Washington, from the rainforests of the Olympic Peninsula to the alpine peaks of the Cascades. Eastern Washington has plenty of opportunities, you might have to wear off a little boot leather to find spots of your own, but they're out there. For deer, the tag is good statewide, you just need to look at the regs and figure out the open dates and specific regs (species, sex and point restriction) for the unit you're in. For elk, you have to decide if you want to hunt Western or Eastern Washington when you buy your tag. Eastern Washington is cow or spike only if you have an OTC tag, branch antler bull is draw only. For Western Washington it depends on the unit you are hunting, some are any elk, some are cow or spike only, some are 3 pt. or better bull only, some are any bull, check the regs. Moose, sheep, goat, and some (if not all) spring bear are draw only. Fall bear and cougar are OTC, though check the unit you're in for cougar as some units are closed. Wolf, grizzly bear, and caribou are strictly off limits. There are tons of coyotes in Eastern Washington so knock on a few doors, someone will give you permission to shoot them. There's also some good field hunting of geese if you can find property that's not leased. My best advice is to read the regs and ask lots of questions if you don't understand something. There is some really good hunting in Washington, you just have to wade through the regulations and deal with a fair bit of competition, do your homework, the animals and space to hunt them are there!
BTW, the archery state record Whitetail used to be held (and may still be) by a guy shooting a self-bow, somewhere out near Spokane!