Then you do not have vertigo, as this is defined as the sensation of the room spinning, even while you are lying down.
The unsteadiness on the feet can come from several different places - the inner ear (which would cause the vertigo), the cerebellum (the part of the brain that controls balance), the actual brain (as fluid buildup can cause similar symptoms, called normal pressure hydrocephalus) and the spinal cord (as decreased sensations to the feet can cause unsteadiness).
Should this become extremely limiting in your ability to walk, you should have your doctor do a physical exam to see if there are clues to where this unsteadiness is coming from, and imaging if necessary.
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M.D. - Neurological Surgery