I'm going to copy n' paste this from another site, since it is a Washington mama and I asked her the same thing. Hope this helps:
"Okay, birth certificate. I'm pretty knowledgeable about this since we had a BC issue. If you get the BC within the first 7 or 10 days after birth (can't remember which), then you are fine. All you do is send dh to the county health department, have him ask them for a BC form. MAKE SURE IT IS THIS LONG FORM WITH LOTS OF BOXES!!! This was our bad part. They gave my dh the wrong form. It was actually the form for ordering a BC (like if I wanted a copy of ***'s) but we didn't realize this, so I filled it out and sent it in. No BC ever arrived. Okay, so I call a few months later to find out what the deal is, discover they have given me the wrong form. I go pick up the right form, when I return it, they inform me that it is too late. I missed the window of opportunity for turning in that form.
So, I had to go to the state records department. Besides looking at me like I was a nutcase for having a UC, they didn't hassle me. I had to do another form, this one we had to get notorized PLUS you have to give them a letter from either your doctor (that saw you pregnant, we didn't have one) or your religious leader (who also saw you pregnant, we also didn't have one) on letterhead stating that you indeed were pregnant and did not steal the baby from someone else. If you cannot provide the letter, you must stand in front of a judge and state that you had a UC and that this child is indeed yours. Now, I'm a little CPS paranoid and didn't really want to inform a judge that I had a UC. So, I called a mw friend who had indeed seen me (socially) and saw that I was pregnant. She was willing to write a letter stating she had "seen" me.
This time around we WILL be getting the correct form. All you have to do on it is put you or your dh's name as the caregiver."
Does that help with your answer?? This mama is from the Puget Sound area, if that makes a difference.