Yes- the stabilizer is not the issue. I would leave it off entirely until you get this resolved. The bracket would have to be bent pretty bad to cause a DW feeling. Can you post any pics- of the control arm, the bracket, and more importantly the bushing(s)? If the control arm(s) is not bent, then you should be able to replace the bushings.
If the bushings are worn, cracked, some small chunks missing- then I doubt that is the problem. If the bushings are completely split, lots of bushing material missing, gaps, etc....then it could be a problem.
This could be anything from a bad pinion angle- meaning that you need to adjust (or upgrade to adjustable) control arms, to an unbalanced wheel. I would look at getting a re-balance, and possible alignment....after checking the other suspension components for any lose bolts (if you haven't checked everything already). Not as likely to be a hub bearing or ball joint issue- with only 12,000 miles- but you may want to check those too.
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