There are excellent exercises to do for balance improvement. Google> vestibular habituation exercises.
After reading a few other posts here, I can agree that once in a while I will sense myself swaying, especially when standing in a line with my legs locked straight. I try to remember to plant my feet apart as wide as my shoulders. In the shower I just keep a rubber mat in there to plant at least one foot on. When I am doing gardening work like lifting things and using my arms a great deal I will plod along and sometimes do the two step side step thing. My husband usually says "I saw that!"
I don't worry about this swaying thing but I realize it is particularly bad when I've spent a long time in a chair doing sitting down activities. I don't exercise on a regular basis as I should.
When a person in a booth behind me in a restaurant jiggles or jerks the seat I get a little dizzy from that disturbance, too. This is my "new normal" as we say.