Tires of the old "ribbed" style will follow grooves. Tires with diagonal tread patterns are much better.
If you angle across the grooves from one side of the lane to the other, it will eliminate most of the wiggle. If it's a couple of miles long like you said, you'll have to go from one side to the other and back again until you're out of the bad zone.
Your instinct to slow down was definately on the money,BTW.
