I would recommend you see a neurologist. Balance problems can be due to a number of medical problems common to people your age. These include spine disease (herniated disk with cord compression), certain vitamin deficiencies (vitamins B12, E, folate, etc.), neuropathy (dysfunction of the nerves, such as the nerves in your feet), strokes (especially in the brainstem and cerebellum), and some specific conditions such as Parkinson disease, progressive supranuclear palsy, and "normal pressure hydrocephalus." There are a host of other conditions that could lead to balance problems. Many of these conditions are treatable; others are progressive but can be treated to slow progression and maximize quality of life.
While many primary care physicians may have a general appreciation for things that can cause balance problems, most generally do not deal with that issue on a regular enough basis to do as thorough a workup as a neurologist might. Given the significant risk of injury and potentially debilitating and even life-threatening conditions that can occur with falls at your age, I would definitely recommend seeing a neurologist for hopefully a more thorough evaluation.