Apparently that T4 test is for Total T4, not Free T4. Only a small portion of Total T4 is not bound to protein that makes it biologically inactive. That small portion is called Free T4. The same applies to Total T3 and Free T3. Free T4 and Free T3 are the biologically active thyroid hormones and they should be tested each time you go in for tests, not Total T4 and Total T3.
So your doctor is not adequately testing you it seems. In addition serum thyroid levels are the sum of both natural thyroid hormone and thyroid med. When starting on thyroid med, the TSH will go down and thus the output of natural thyroid hormone will also go down. The net effect is little/no change. Only when TSH is suppressed enough to no longer stimulate natural thyroid hormone production, will serum thyroid levels reflect further increases in thyroid medication. So you need to keep increasing your med dosage until your Free T4 and Free T3 levels rise enough to relieve hypo symptoms. Typically that requires Free T4 at mid-range, and Free T3 in the upper third of its range, or as needed to relieve symptoms. The FT4 and FT3 reference ranges are far too broad to be functional across their entire breadth for everyone. So a good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting FT4 and FT3 as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results.
I don't recall hearing of the pitch problem you mentioned. So if Hashimoto's Thyroiditis is the possible cause for you being hypothyroid, it would be good to get an ultrasound of your thyroid gland to check for nodules/inflammation.
In addition, hypothyroid patients are frequently deficient in Vitamin D, B12 and ferritin, so it would be good to get those tested and then supplement as needed to optimize. D should be at least 50 ng/mL, B12 in the upper end of its range, and ferritin should be at least 100. Those can have a significant effect on fatigue also.
My voice was deeper and kind of rougher sounding with untreated hypothyroidism. My voice change was due to mucin build up thickening the vocal cords. Voice changes can also occur with thyroid masses or thyroid nodules that stretch the laryngeal nerve.