With Hashi;'s confirmed by the TPO ab test result, along with the high TSH and your FT4 and FT3 results, and you feeling awful, and with the advent of iodized salt, why are you still convinced that iodine is the underlying cause of your problem?
seems to me that your first priority has to be to get your Free T4 to the middle of its range, at minimum, and your Free T3 in the upper third of its range, as necessary to relieve hypo symptoms. Since hypo patients are so frequently low in the ranges for Vitamin D, B12 and ferritin, those should be tested and supplemented as needed. There are a few other tests that may become advisable, but those would be a good start.
By the way you don't need to fast for a thyroid test. If you are taking a T3 med, then it is best to defer your morning dose until after the blood draw.