Most doctors don't understand that TSH is useful as a diagnostic for thyroid status only when it is abnormally high. Thus it is useful only for identifying overt primary hypothyroidism. Hashimoto's Thyroiditis falls into that category. When Hashi's is suspected, the tests to confirm is TPO ab and if that is negative, then a test for Thyroglobulin antibodies (TG ab) should also be done.
Because of your high TSH and symptoms, your doctor decided to start you on T4 med. Most doctors don't understand that T4 med is not additive to your prior level. This is because the med causes TSH to drop, thus reducing stimulation of the thyroid gland and reducing output of natural thyroid. Your serum levels of thyroid are the sum of both natural thyroid and thyroid med. Only when your TSH is essentially suppressed will further increases in your med start to raise your serum thyroid levels. Your 75 mcg of T4 is not nearly enough, as evidenced by your relatively low FT4.
In addition you need to test for Free T3 since it is FT3 that is most important since it causes the metabolic activity in your body. T4 is only a prohormone, available for conversion to T3. Unfortunately doctors also don't understand that most hypothyroid patients taking T4 med don't adequately convert T4 to T3, resulting in FT3 levels that are too low, and continuing hypothyroid symptoms. So, many hypothyroid patients benefit from the addition of T3 to their med.
I don't think you need a Rheumatologist at this time. Based on my own experience, lack of FT3 can cause symptoms like you describe Also, you should be aware that most Endos rely only on TSH to diagnose and treat, which doesn't work for most hypo patients. What you need is a good thyroid doctor. By that I mean one that will diagnose and treat based mainly on symptoms, along with both FT3 and FT4 levels. To better understand all this I suggest you read a paper I co-authored. Then come back and ask any additional questions.