Welcome in H.C.M.
I have gone through your query regarding sharp pain
near Adams apple on gentle palpation during swallowing. You are in doubt as you had damage thyroid cartilage
Here I would like to tell you that you had palpated gently over thyroid region so no question arises for fracture of cartilage. I think most likely you are having some inflammation of pharynx and that is creating this pinching pain
To confirm it a thorough examination of pharynx including endoscopy
of throat is necessary to pin point the diagnosis.
I think your query is answered.