I should also disclose I was formerly hyperthyroid and took radioactive iodine, I am now recently diagnosed hypothyroid (high TSH levels) but am awaiting results for free T3 before beginning any medication.
Im gonna touch on this a bit but you have already caused yourself some major thyroid damage. Do you understand why radioactive iodine is used?
The thyroid basically has iodine receptors, iodine is the main elemental building block of thyroid hormones. When someone has a hyper thyroid they give them radioactive iodine because they know it will be directly absorbed by the thyroid and in turn the radiation will damage the thyroid reducing its output.
This is akin to amputating your arm because you have a broken wrist. Its barbaric medicine at ts finest.
Now you say you have a high TSH and that your doc says you are hypothyroid (ie underfunctioning thyroid). Do you know why that is? Because the radiation damaged your thyroid and in its damaged state your elemental (ie non radioactive) iodine levels are low so combined your thyroid simply can not do its job.
You and your doc both likely dont realize that TSH is a poor indicator of thyroid function. TSH is not just a measure of the thyroid trying to produce more T3, TSH is also released to stimulate sodium-iodide symporter cells. Basically TSH is released to cause new iodine receptors to be created so the bodies iodine can be gathered and converted to T3. Damage your thyroid with radioactive iodine and you damage the cells that have symporters, of course your body is going to try and recover from that damage by increasing TSH to cause more symporters to be made.
Id bet big money your problems all stemmed from iodine/selenium deficiency to begin with and they will never clear up until you become sufficient in those nutrients. No amount of medication will fix your thyroid but supplements can.
You can blindly follow your doc and simply mask symptoms for a while or you can try and fix yourself. Do yourself a favor and get the book "Iodine: why you need it" and save yourself a lot of money and misery.