After starting on thyroid meds, TSH is even less useful as an indicator of thyroid status. The most important indicators are symptoms and also levels of the biologically active thyroid hormones, Free T3 and Free T4. Free T3 is the most important because it largely regulates metabolism and many other body functions. Scientific studies have shown it to correlate best with hypo symptoms, while Free T4 and TSH did not correlate at all.
You really do need to test for Free T3, along with Free T4, in order to see if you are converting the T4 to T3 adequately. I would also suggest that if not done previously, you should test for Vitamin A, D, B12, and ferritin.
When you talk to the ND keep in mind that a good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T3 and Free T4 as necessary to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not just test results.