I started taking the levothyroxine for a hyper thyroid about the time when the nasal swelling started. My research as indicated that there are side effects to the Levothyroxine.
You describe “considerable sinus vocal chords and upper lung inflammation”. I am uncertain about what this statement means but assume the symptoms to be those of a cold or another type of upper respiratory infection. Such symptoms may be caused by persistent or recurrent infection but, in many instances are caused by repeated exposure to, 1) an allergic substance or, 2) to an irritant; that is, they can be caused by something present in your environment. With that in mind it might be helpful for you to take a careful inventory of any and all substances in your environment that might fit into one of those categories, especially those just introduced a little more than two months ago.
You also say, “when I stop the swelling is considerable.” I assume you are referring to swelling of the thyroid gland and that would not be surprising. Nor would voice change be unusual when your body is low on thyroid hormone. The lack of that hormone can result in changes in multiple parts of the body including skin, intestinal function, heart and lung function, anemia and most importantly a lack of energy and generalized slowing of body functions. It can also cause swelling of the vocal cords resulting in a change of voice.
If you truly have low thyroid function, the most important thing now is to determine the cause of it and get it effectively treated with replacement therapy. Once that is accomplished, starting with low dose replacement therapy, you may find that your other symptoms resolve as well.
Having “no medical” is a major problem, but I urge you to find a way to get the thyroid condition evaluated and treated.
Good luck