Yes, I'm afraid it's legal. The rule falls under the exception of public/private roadway of RCW 46.61.005. Thankfully, RCW 46.61.667(6) specifically states that it will not go on your driving records. You might be able to mitigate the fine if you don't have any prior traffic violations.
I agree with the other attorneys, the statute of limitation is 3 years in Washington, not 2 years. From the medical bill amount it appears you and your interest may be best served by hiring an personal injury attorney who will be able to guide you through the process. Lawsuit is quite complex and the insurance company will make sure to make it hard on you if you are not represented.