If you are not assimmilating nutrients, its more likely to be a digestive problem, than one of Iodine
deficiency. I had this problem when I had low stomach acid.
Puffy eys, dry skin, and hair loss, are all symptoms of hypothyroidism, but weight loss and greasy hair, are symptoms of hyperthyroidism. All these symptoms could also be from mineral and vitamin deficiency as well.
Iodine supplementation can help with all the symptoms you are experiencing. Including helping with stomach acid production. Also bitters, and nettle leaf, will help with digestion as well. I also found that taking vitamins and mineral supplements in liquid form, is better for absorption, rather than capsules. (If you have low stomach acid, you will not be able to break down the capsules).
Green smoothies, fresh juices, and soups, are also much easier to digest, if stomach acid production is an issue.
Have you had your thyroid hormones checked?
Or have you done a basal body temperature chart?