I would feel terrible if I was taking as much iodine as you are and that is a plain fact. That amount for you may be way too much of a detox for your body to handle right now, or you may simply process it differently.
My brother in law has hep c, and guess what? The oil pulling, for a mere five mins made him violently ill for two days (and he never had any desire to return to it). If you have a toxic liver you WILL feel the brunt of the detoxification process doing something that makes many feel energetic without receiving the immediate noticable benefits others are touting. It would make perfect sense that to do large amounts of iodine if you already have this type of congestion would leave you tired and feeling like crap, and yes, the more of a toxic load you lay on the liver can definitely affect your energy levels and thyroid. Cutting back to one-two drops a day, increasing your water, salt, c, and maybe looking into liver and kidney support would be a better plan than jumping on meds. I am not sure if you took off regular time on the iodine each week or how long you have been on the iodine but taking a few days off a week or spacing your doses to every other day might help your energy levels. Be sure to keep on the cleansing aspect of the program even on your days 'off' from the iodine...your body will still be releasing toxins. Look into those footpads, they really helped me immensely. You can just get 10 nights worth to see if they help you. I am not sure what your history is, but it sure sounds like toxic overload to me.
Hate to see you despairing so, and the thyroid med is no quick fix if that is what you are hoping for. You might find a quicker result simply by focusing on helping your body move out the toxic load.