The concepts of swing and sway defy easy simplification.
Generally, swing is forward and backward with respect to the feet, but rapidly becomes sideways in the body as the CBM rotation happens early in the the step where swing occurs. Those who insist on delaying their CBM past the proper point will have a more forward-backward swing in the body, but they will still end the swing with the body fairly sideways to it.
Sway is the position which results from the combination of swing and CBM - if your swing ends with the body moving sideways, then one side will be higher than the other and we call this sway. You can also fake a similar position without swing, and some people would call that sway as well, but it's a decorative shape (or used to aid static lines) not one involved in movement across the floor.
In actuality you can have sway without CBM, if you start your swing with the same side lead you end it with, you don't really need CBM to achieve sway. But usually we use CBM to change leading sides during the course of a swing. If you executed a swing that was actually forward or backward with respect to your body (not something you want to do) and ended up with an incline forward or backward rather than at least diagonally sideways, it's unclear if this could be called sway, but it's not really a usefull question as it's not a position compatible with partner dancing.