I'm on a diet low in sugar and regularly I have dizziness, which can be?

Anxiety. I assume your diet is a result of fear of overweight and fear of diabetes. You are reacting with a stress response. Be sure it is not low blood sugar or high blood pressure. Get a thorough medical evaluation.
What are you eating? A low sugar diet is healthy. Your body doesn't need simple sugars, it can extract them from other foods. Be sure you are eating regularly and getting enough protein, fruits, and vegetables.