Try the steel wool first then tumble. I am not talking about buffing them with wool till they shine just hit the bad spots and spin the case inside the wool.
I just cleaned some 300 win mag that were 20 years old and they were nasty as heck. Some of em green. I added a few cap full of car buffing compound because I am out of case polish. 40 min later they look like new. Just make sure they do not have any compound left on them especially near the primer pocket.