The result of Hashi's is a person becoming Hypothyroid. Which means low thyroid as the antibodies attack the thyroid gland and thus the gland produces less and less thyroid hormone leaving the person deficient in the amount of thyroid.
One symptom some people have of being low thyroid (hypo) is dizziness. So it would not be unheard of for a person who has Hashi's to also experience diziness.
Please post your thyroid related test results and their reference ranges shown on the lab report. Also, what thyroid med are you taking and what daily dosage? Do you have any other symptoms besides the dizziness?