One of the requirements to be legally able to marry one person is that the prospective spouse is not currently married to another person.
You can look at the marriage application to see whether you would be certifying to something that is not true if you sign and submit the application.
Looking at the WA marriage license application, I do not see any statement that the applicant is certifying that the applicant is currently married. Since the applicant is not certifying that he is not married, I suppose that some argument can be made that the applicant is not falsely certifying anything or violating any criminal law by merely filing the application and does not get married before his divorce is final.
You can check with an attorney in your state who can review the case law of your state to see if you can "file for a marriage license if you are not divorced yet".
One the other hand, it may just be much cheaper and safer to wait until the divorce is finalized before doing anything.