A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free
T4 and Free
T3 levels as needed to relieve symptoms, without being constrained by resultant TSH. However, you may well need to address your low cortisol before even starting on thyroid med.
I would suggest that you get tested for Both Free T4 and Free T3 each time you go in for tests. Also, you should get tested for Vitamin D, B12, ferritin, and DHEA.