I would at most use the scotch brite pads. Maybe try the blue ones first, as they won't scratch the metal.
What kind of grease are you using on your slides? Also, have you tried cleaning your valve casings with a conventional casing brush?
There are some nice "low brass care" kits out there, which will have a rubber coated snake. That can really help get the gunk out of the inside of the slides.
Do you have conventional water keys, or something more "advanced" like amado keys?
After your next cleaning, I would consider getting some BRASS SAVER plastic snakes, and using those to clean out the valve slides once a month at least.
You can also simply take off the slide, put some water inside, cover the end with your hand, and SHAKE. That can help dislodge any crud/goo/etc that may be in there.
Sterling / Perantucci 1065HGS Euphonium, 1952 B&H Imperial Eb Tuba, and a bunch of trombones.