A big set of car keys, some vinegar and "never dull" which is basically kerosene and wool batting used to polish metal. Drop the keys on the covers a few times. Tarnish gently with vinegar. Polish a bit.
I think the key to string shadows isn't to tarnish under the strings so much as to reshine where the strings aren't. Little light polishing parallel to and OVER a few sets of strings right before string change works. Then play and play. I've probably played 20+ hours on these pickups.
I don't really use anything as abrasive as sandpaper or steel wool. I get better results by polishing.