Here in Colorado as well. I admit a batch of 45 Colt went in the oven to dry off this evening, Convection bake at 200F for 10 mins
I use food grade citric acid, 5 lbs bags usually ship to the house for $15 -$20 dollars. I use two table spoons and s good dollop of washing up liquid with a quart of hot water. Lasts me a long time. Never more than 5 mins, generally 3 mins in the solution. Stir the brass, empty the bucket, rinse with cool water. Brass is super clean inside and out - primer pockets too, as you note. The citric acid also passivates the brass. It'll stay tarnish free on the shelf for a very long time.