What causes under 20 mph side to side sway when the steering wheel is not wobbling?

What you're describing is typically caused by either a bent wheel or a "slipped belt" on one or more tires.

Often a radial tire belt can become detached from the radial plies, allowing the belt to slip to one side. When that occurs you'll frequently hear a squeaking or crunching sound from the tire as you allow the vehicle to move slowly.

A slipped belt is not safe and should be dealt with as soon as possible.