Don't under estimate the knowledge of an ENT. My ENT was the one who got my Hashimoto's diagnosed because my pcp refused to help me - based on TSH levels......... My ENT was the one who referred me to an endo when my pcp refused........
A lot of endocrinologists specialize in diabetes, but know very little about thyroid, even though it's also an endocrinology issue.......
Pick the one who is willing to test for TSH, FT3, FT4 and antibodies - that's the one who will most likely help you the most.........
An endocrinologist would be my choice. I have had thyroid issues since I was 18 and I am now 36. An endocrinologist deals specifically with hormonal related diseases, such as hypo or hyperthyroid, diabetes, etc. An ENT doctor doesn't. I don't know how much an ENT doctor can do for you, but, in my opinion, they would deal more with breathing issues, tubes in the ears, snoring, sleep apnea, etc. Good luck.