There is a paid leave Oregon program that starts next year, but I think nothing now other than any sick time and short term disability from your current employer. You probably need to apply for your time off under federal FMLA to protect your job, but that is just time off guarantee and no pay.

WA PFMLA requires that you have paid in through worked hours in Washington. If the premium is not coming out of your check then you will not qualify (min 820 hours worked and paid in to qualify). have you had any WA state pay since the program started?

ETA: If you are an independent contractor, there is a way to pay your own premiums to join the program, but You would know if you were paying the premium then.