I just graduated from Seattle University in June but took my time studying for the NCLEX and have just started to seriously look at jobs in Seattle again and right now they are a bit sparse but still out there. Let's just say I graduated in a cohort of 62 and I would say well over half of them have gotten jobs in the Seattle area within practically every hospital. Alright so brain drain on Seattle hospitals:
Seattle Children's is a great choice if you are wanting a peds position. Just a heads up that they post new grad positions on Wednesdays. G4 or G3 units post a lot of new grad positions, those are both surgical units. The medical unit is another area that usually posts quite a few positions (makes sense as I believe that there are three or four floors and 125-150+ medical nurses at the facility). While it may be hard to get your foot in the door, if you did your senior practicum on a pediatric unit you have a good chance of getting in, they take nurses from everywhere
Harborview and UW Medical Center:
Not known for it's L&D or Pediatric units really, BUT they both are known for hiring a lot of new graduates. I loved my clinical at Harborview, but I have never stepped foot in UWMC, they have great Med/Surg units at both hospitals.
Swedish Medical Center:
Swedish is great for L&D/Postpartum etc. they also post individual positions for new grads (at least that is what I have seen). I did my maternity rotation at Swedish and I have to say that they lived up to the reputation of popping out a lot of babies , if you prefer a more natural approach to birthing I would definitely look into Group Health (though I don't know of anyone that has gotten or even tried to get a new grad position there so it's really not on my radar).
Valley Medical Center:
The BEST professor I ever had at SU was our maternity professor, this is the hospital that she works at. It is a great facility and has a really strong set of nurses. One of my close friends did her maternity rotation there and said it was fantastic. The nurses are caring and make a ton of extra effort to make the birthing process a good one and the birthing center is really nice.
That's all I can really think of now. I haven't looked at Franciscans at all so can't really speak to its L&D reputation.
Best of luck! And congrats on graduating with your BSN!! It has been a tough four years of classes and clinicals, I hope you get a job soon!!