"Leash laws" are generally instituted and enforced by a city or a municipality. There is not a statewide leash law in Washington.
There is a Seattle ordinance that requires all pets (except for cats and, strangely enough, pigeons) to be kept on leashes anytime they are off their owners' property. The pet may be "at large," which means that it can move around without a leash, in certain parks, commonly referred to as "dog parks." This dog park rule wasn't added until 1997.