Resonance Problem

how long do you keep this cartridge?
if this is a suspension problem which is most-likely, you should increase the tracking force by a couple of fractions or even by half-gram.
less-likely grado will cause some mismatch with the tonearm you're using right now.

As far as I know, 12g qualifies as a medium-mass arm, and since it falls within the range that you say Grado recommends for this cart, this doesn't seem like a mis-match. I don't know how you are estimating or measuring the frequency of the observed motion of your woofers, but a cart/arm resonant point in between 10-20Hz is generally considered desirable, because it occurs below the frequency of the bass content on a record, but above the frequency of most warps. Although I'm unfamiliar with your 'table/arm, I'm suspecting some kind of acoustical or mechanical feedback problem instead. You say you can "see" the problematic motion looking at the cartridge, not just at the woofers. Is this true when the electronics are off, or the volume is turned all the way down? If you can't see the cart motion in that situation, then you probably need to try a more isolated mounting scheme for your TT, or move it farther away from your speakers, or both. (Even if you can't see a difference in what you think is the cart motion, it is worth trying these approaches, just to be sure and rule out a simple explanation.) However, if the TT has some sort of tuneable suspension setup, you are going to want to call on that manufacturer or dealer for advice on optimizing that, as well.

I had a similar problem and solved it with tube traps, moving the rack farther from the speakers, and putting a Big Rock isolation platform under my turntable. Nothing wrong with my cartridge.

Have you tried moving the rack position in the room?
Can you find if the effect is present with low fr. music playing?
There´s a possibility that you have your rack in an accoustic reinforcing point of your room.