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There are several methodone treatment centers in Washington, but I don't know how close they are to Silverdale. Here is the web site for one company: http://www.evergreentreatment.org/location.htm
There has been much debate about opening a clinic in Bremerton, which I do know is in Kitsap County not too far from Silverdale, but the neighbors have managed to drag out the zoning for at least the past 3 years. As of today it still doesn't exist.
According to the Kitsap County Health Department, the nearest methadone treatment facility is in Lakewood:
Tacoma Treatment Solutions
Lakewood, WA 98499
There is another in Pierce Co.
Tacoma Pierce Count Health Department
3629 South D Street
Tacoma, WA 98418
You would need to contact them to see if they can get an order from your Florida provider to dispense treatment while you are in Washington.