I’ve tried several different methods of pre-washing or not and storing berries but the one that works the best for me is a few minutes of soaking in cold water with a few tablespoons of white vinegar (confession: I totally eyeball it). I give them a gentle rub with my fingers.

After they have soaked, I lay them on a towel to dry. They must dry COMPLETELY before storing them in the fridge.

I typically store them in an open bowl so moisture isn’t trapped in. I sometimes will place a washcloth or paper towel in the bottom and cover with a kitchen towel just to catch any extra moisture. My kids don’t let berries last very long. They will devour 2 pounds in a day, easy (I have 3 kids).
I do the same for grapes

You can also use baking side instead of vinegar to help remove residue. I found vinegar is great to prevent berries from molding too quickly. Have you tried this method? What works for you?