Under the new Washington Family Leave Insurance law, an employee who has worked 680 hour during the past 12 months for their employer is entitled to paid leave associated with caring for a new child. That's really the only substantive requirement.
Unemployment, on the other hand, requires that the claimant must be able and available for full time work. Consequently, you cannot quit work because of a disability, and receive unemployment benefits.
Note: Washington's new law took effect in October, and from what I can discern, not even the government completely understands what is lawful. There are reports from various nongovernmental sources claiming that if you are disabled due to pregnancy, you can start receiving family leave benefits. However, I find nothing in the actual law or state employment regulations to confirm these reports. Therefore, I suggest that you try to conform your circumstances to what is actually found in the law, which you can review for yourself at this link.
I realize that my answer may not be exactly what you were hoping to read. However, under the circumstances, the best that I can do is to explain what the law is and is not, so that you can avoid expending valuable resources looking for answers that do not exist, and concentrate on the options that are actually available.
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