One of my answers was When driving, swerving is to avoid something, weaving is to go in and out of traffic. So shouldn't the saying be "Weaving All Over The Road," instead of "Swerving all Over The Road."?
In this case, swerving all over the road is just an expression designed to describe a person's reckless driving. You could use either saying but swerving indicates a more uncontrolled and reckless situation.