About Washington State Carry Permits:
They must issue you one if your record is clear, you submit the required fingerprints, and you pay the fees. Oh, yeah...and you have to have been a state resident for (I believe) one year.
The "which PD to go to" question, therefore, hinges upon which one will give you an assured appointment to take your fingerprints (because the fees are all the same).
Up here in the islands, permit fingerprints are taken only one day a week. Walk-ins are OK, but they prefer to know that you're coming. They don't do the job during lunch time.
That's all I know.