More than likely what's bothering your teacher and your wife is the quality rather than the quantity of sway. So instead of avoiding it altogether, I would suggest focusing your attention on how to sway correctly. Soft, "noodly" sway can be uncomfortable and irritating in the smallest of quantities. But when done well, you can actually get away with doing quite a lot.
If you are indeed noodly, if you tend to "break your sides", or worse, involve your arms in your sway, you'd be well served by learning how to brace the upper body and use base sway. Not all sway ultimately comes from the base, but since most beginners use no base sway at all, and therefore overcompensate with exaggerated top action, the first step toward overcoming the problem is to take the top sway out of the equation entirely. Take the time to learn how to sway strictly from the base using knees & ankles, then add top sway much later once the base has been mastered.
As far as top sway is concerned, when you do finally get to it, you can improve it with any of various exercises aimed at improving strength of core, lifting the ribcage off the hips, and maintaining good posture in general. And though it may sound cliche, it is nonetheless important to remember the age old principle that top sway must be produced by stretching one side without collapsing the opposite. To be sure you're following this rule, stop any any time while in a sway position, place your hand on the low side by touching the fingers on the hip and the thumb on the rib cage, then stretch them apart. This corrects the collapsed side and teaches the body how to achieve top sway without it.
Regards,
Jonathan Atkinson
www.ballroomdancers.com