There are some pretty good shooters over in the Spokane area - Randall VanZant for one. Ron Sanders or Praveen Kumar (I think he shoots over there from time to time), and Dean Brown are all good guys and will steer you straight.
If you're closer to Wenatchee, give me a holler.
And down at the Tri-Cities, they have the (in)famous Rattlesnake Mountain Shooting Facility (www.tcsa.info). The Long Range matches might be a bit much for your gear at the moment, but they have some pretty active Friday shoots @ 300/600 yds. George Crawford, Bill Prue, John Hauptmann and their crew will get you all lined out.
I'm in Pullman, so Spokane is definitely closer than Tri-Cities. I'm less than 8 miles from the Idaho border, so Moscow, Lewiston, Couer d'Alene, and surrounding areas are also on the table. Considering how many people I know in Pullman who love to shoot (students, farmers, scientists, etc.), it blows my mind that there isn't a single rifle range within 25 minutes in zero traffic.
Just out of curiosity, is it just bench rest shooters who clean their bore between shots, or is that standard practice for the other competitions as well?