Has it been taken apart? You might try taking it apart and making sure that the shaft is pressed down as far as it can be. If it was put back together somehow without that shaft pushed all the way down into place, it might be wobbly.
While you're at it, take apart one of the faucets that's working well and compare. Notice that there's also a little plastic piece that channels the shaft so that it moves only forward and back, not left and right. That might also be missing for some reason on the faucet, or perhaps isn't working for the reason mentioned above.
Good luck!