My Cheng Shin's wobble a tiny bit sometimes on grooved pavement. I try to just relax and let it wander a bit. The tires do have a small center groove but they are much better than the Dunlop's that were on my GZ. The metal bridge gratings freak me out sometimes with the wiggle waggle business but if I just keep a light hold and slow down a bit it wiggles less.
But back to those winds...a real pain I am sure. Especially with crap flying around. And the cages and trucks being pushed around as well, no picnic!