Free: Contests & Raffles.
WA or any other state, in my opinion there is no OTC opportunity - or even a good odds NR draw - for a probability of killing a mature bull on public land without a significant investment in sweat and money. Good luck! You are the perfect age to start on what should reasonably be viewed as a 3-5 year quest.
Are you hunting archery? Should be several decent areas on the westside. Call WDFW ask for the big game bio in charge of the area you are hunting and start asking questions.