Since you have hypothyroidism, do you know if you have Hashimoto's?
Have you ever had a thyroid ultrasound to determine whether you have nodules on your thyroid or not? Many of us with hypothyroidism/Hashimoto's have nodules on our thyroid. Typically, they are nothing to worry about, but sometimes, the nodules can swell up. Sometimes the thyroid, itself can swell up/become inflamed... this is called a goiter. Both, nodules and goiter can also cause the feeling you refer to. I get it periodically, too.
While it's possible for nodules to be cancerous, it's important to know that less than 5% of all nodules are cancer, so it's pretty rare.
The problem it is that the thyroid is located near the bottom of the throat, so I'm wondering if your problem could be tonsils or something else.
Do you take replacement thyroid hormones for your hypothyroidism? If so, what medication(s)/dosage(s) do you take and how long have you been on it/them? If you have a copy of your latest thyroid hormone test results, it would be good if you could post them so we could see your actual status. Be sure to include reference ranges with any results you post, since ranges vary from lab to lab and have to come from your own report.