Hello and welcome to ‘Ask A Doctor’ service. I have reviewed your query and here is my advice.
A swelling in the Adam's apple zone that doesn 't regress with antiinflamatory drugs may be a local infection but it is necessaty to do some test to confirm the infection.
If the cause is an inflammation of the thyroid gland or subacute/acute thyroiditis
than you should have tenderness or pain in the thyroid apart the other local signs of inflammatin (swelling,redness, fever)
The infected tooth can also produce the signs of local inflammation in the respective zone near the tooth.
About your question,i suggest to discuss with your dentist
if the actual swelling in the Adam's apple zone is the the result of a potential infected tooth in the jaw.
If there is infection the antibiotics may help.
I can say that usualy hypothyroidism
do not cause simptoms like yours.
If you have local pain or tenderness i suggest to consult physicaly a local endocrinologist
because it is necessary to do a thyroid ultrasound and some blood test
to see the actual function of the thyroid gland.
Hope I have answered your query. Let me know if I can assist you further.