I had an accident in a car several years ago where the car slammed into a
curb sideways and failed the control arm on that side.  I got the car back
from repair and all seemed fine for several weeks until the first time I
drove it in rain.  Then it would sway from side to side as though I were
driving in the ruts left by studded tires except there were no ruts.  To
make a long story short, it turned out to be that one of the trailing arms
on the rear wheel had also been damaged so that the rear wheels were 'towed
out' pretty severely.  On dry pavement the tires would just scrub but if the
surface were a little slick the car would chose one side to follow with the
other side doing all the scrubbing.  This would cause the car to turn toward
the scrubbing tire and when I corrected with the steering the weight would
shift and the car would start following that wheel.  So I would go down the
road like a drunken sailor -- very disconcerting.

Have a 4 wheel alignment check done.

Bad rear subframe bushings can make the car wander a lot when changing
direction. And at your age you're about due for a new set.

If you've "upgraded" the car with wheels larger than 16" or tyres wider
than 225mm this won't help either.