You need to see a good endocrinologist. You don't mention your lab results. Is your doctor checking Free T4 and Free T3? When were they last done and what were the results? Have you had any other studies of your thyroid gland done?
Sounds like your doctor is following the usual procedure for treating a hypothyroid patient. Thyroid med (usually only T4) is adjusted as needed to get a TSH level makes the doctor happy. That doesn't work for most people. Based on what you have said you are very under medicated. If you want to understand why, I suggest that you read my paper posted on the Thyroid UK site.
https://thyroiduk.org/further-reading/managing-the-total-thyroid-process/
From that you will find out that thyroid med should be adjusted as needed to relieve hypothyroid symptoms, without going so far as to create hyperthyroid symptoms. Usually this requires Free T4 to be at least mid-range, and Free T3 to be in the upper half of the range, and adjusted from there as needed to relieve symptoms. In addition, it very important that Vitamin d should be at least 50, B12 in the upper part of its range, and ferritin should be at least 100. Having adequate thyroid levels and also ferritin ( a form of iron) is very important for good hair growth.
You don't necessarily need an Endocrinologist. They are usually too rigid in their beliefs to adequately treat a hypo patient. What you need is a good thyroid doctor that will treat clinically, as needed to relieve symptoms. If you will tell us your general location, perhaps a member can suggest a good thyroid doctor for you.