Unsteadiness as a symptom has multiple underlying mechanisms for its causation. One of the common symptoms is of vertigo, which is in fact a sensation when the entire environment in and around a person seems to be moving / reeling / spinning, and the person has the feeling of being drawn into it. From your history, it appears that you don't have vertiginous sensations.
The other systems that cause such feelings are cerebellar (two little brains that hang below the big brains called cerebrum) involvement and occur primarily due to in-coordination.
At times, sensory nerves or the sensory tracts of spinal cord can get involved in a variety of diseases, which may again produce the symptom of unsteadiness. This is particularly common in our country due to vitamin B12 deficiency in strict vegetarians.
Finally, generalised deficiency states like moderate to severe anaemia can cause a sense of fatigue and unsteadiness at times.
I suggest that a consultation with a good neurologist may be rewarding.