You aren't allergic to shellfish, seaweed, or CT contrast dye are you? Thyroxine contains a tiny amount of iodine per pill, not enough to cause problems in normal doses. You'd have to take a LOT of thyroxine and be VERY allergic to develop a severe allergic reaction, although I guess it's possible. Do you have hives? Itchy, swollen spots in and around your mouth?
I had a severe allergic reaction a couple of times, and I knew I wasn't imagining it because my tongue was so swollen that I sounded strange enough for others to notice when I talked. Also my tongue was itchy. You attract attention when you scratch your tongue.
More commonly, tightness at the back of the throat is caused by anxiety or acid reflux. Sometimes Benedryl (diphenhydramine) can help both anxiety and allergies. In an emergency, buy liquid-gel packets, snip off the end of the Benedryl.
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Thyroxine Poisoning