I have been an asthmatic my whole life, could the medicine cause hypothyroidism?

Iodides can: Cough medications or expectorants,if taken in excess for years can overexpose the thyroid gland to iodides which can ultimately shut down the thyroid gland and cause it to be underactive. This is a rather uncommon situation however.

Hypothyroidism : Unusual causes of drug-induced hypothyroidism include: Amiodarone Eating a large amount of iodine-containing seaweed Nitroprusside Perchlorate Povidone iodine (Betadine) Sulfonylureas Drugs which used to treat hyperthyroidism could do it ,too like PTU, (propylthiouracil) radioactive iodine, potassium iodide, and methimazol. Hypothyroidism is not uncommon among women.