Welcome, in advance!

I've not rented for a number of years, but I would google search "Seattle Apartment Rentals" first.

I'm not sure what living conditions you're used to in New York, but having a large dog can make apartment hunting problematic.

If you can get a job in place first, you'll be much better off.

And public transportation? Nonexistant. Possibly the worst in the country.

No, I'm not kidding.

For locations, centrally located neighborhoods would be Downtown, South downtown (SODO), Capitol Hill, Queen Anne (pricey!) Ballard, Wallingford and the University district. If you can afford somewhere in that area, a bike will do you well for commuting.