For your best chance at taking an elk, head to ID, MT or CO. You will have a chance at more animals, more liberal seasons, less pressure and less limitations.
In MT, you can hunt elk for something like 11 weeks on a general tag and in many units you can shoot a cow or branch bull, some are any elk. The ratio of hunters:elk is just so much better one you cross the border.
In WA for rifle, you may be looking at a spike only unit during a very limited season. What percentage of the elk in the unit are spikes, 10%, 5%? Compare those odds to a unit where there are more elk and you can shoot all antlerless plus everything with a brow tine. Your odds just went up probably 10 fold.