Hello; I am an Audiologist and a long-time user of corrective lenses.
The disorientation that can happen following a change in prescription is typically related to depth preception, such as missing the step or being able to catch a ball as @kyanblue describes. Also, when you are moving, the curvature of the lens sends a distorted image to the brain. I once had the eyeglass lab put my prescription in backwards (my Right eye is much worse than my Left) and that really had me off-kilter! They did correct the error. It couldn’t hurt to have the prescription of your lenses double-checked.
However, you say that you feel as though you are swaying – even when you are sitting still. Does it occur even when the glasses are off or your eyes are closed? Ocular influences on our equilibrium typical manifest as an inconsistency with how we are relating to the space around us. But if you feel as though you are moving, that is more often an inner ear or neurological factor of the complex balance system.
This time of the year is when many people experience dysequilibrium, and it seems to relate to the change in the seasons, allergies, and pressure changes in the sinuses and ears. Dehydration really exacerbated this, as do fatigue/lack of sleep, and poor nutrition; assuming by your name that you are a female, your cyclic hormonal changes can also play a part.
So as with most minor malaise, the first recommendation is to simply take better care of yourself… take it easy, drink lots of fluids that do not have caffeine or alcohol in them, and eat nutritionally balanced meals. If you have medications that you take, be sure to take them consistently as directed. If this feeling persists after a few days, see your Doctor.
See a Doctor as soon as you can if you get a more intense spinning sensation with nausea, ringing in the ear(s), changes in hearing or vision (with or without the glasses on), or headaches. I hope you feel better soon!