Not sure on the hypo/hyper swing. That's a possibility with the Hashi's. Regardless, you'll want to up your selenium. Get some selenomethionine and take 400-800 mcg a day.
As for the magnesium, if you feel you are having absorption issues, try transdermal such as Ancient Minerals (mag oil for the skin).
The selenium, magnesium, and other antioxidants are critical to bring down your antibody levels. Also, thryoid meds like Naturethroid can help stop the "burn" of the Hashi's on your thyroid. Add some E for more antioxidant activity.
Your symptoms also could be adrenals. Chelating is very hard on the adrenals as is being hypothyroid. Get some Isocort and take it in the morning with your iodine. I take 3 in the morning and 1 four hours later if I feel like I need it. You can take up to 8 a day. For some people with adrenal issues, the Isocort is too weak, and they need hydrocortisone.
Also, listen to Trapper :) And take more unrefined salt throughout the day if you feel like you are detoxing.
And one more thing. You'll need the B vitamins that are part of the Iodine protocol. I take ATP Cofactors plus a complex.