Check out some of the local shelters via these links:
PAWS cat adoptions
MEOW cat adoptions
Seattle Animal Shelter Cats
Many kitties choose their own person. What to do if a stray kitty shows up at your home or work place? If the kitty does not have a collar, take him or her to any vet clinic and ask to have the kitty scanned for a microchip. This is free and you generally don't need an appointment since it only takes a few minutes with a technician. If no chip, take the kitty home and make him or comfortable but ask around your neighborhood and look for posters of lost cats. Many neighborhoods have emails lists or FB pages that work well for information exchange of lost and found animals. If no one appears to be searching for the kitty then warm to the idea to you may well be adopted by this kitty. A detailed veterinary exam is probably the next step. Fortunately there are lots of low cost spay/neuter options around if you rather use the extra cash to buy a luxurious kitty condo, plush beds and loads of premium food to spoil the poor kitty who was wandering around for who knows how long looking for someone like you.
Adopt a cat or let a cat adopt you, either way this is one long term relationship that will always be sweet and puurfect.