Since you are taking
levothyroxine,
and not being tested for Free T3
and Free T4, I suggest that your first step should be to get that testing done, in order to find out your actual levels. If the doctor resists, just insist on it and don't take no for an answer. It is that important.
Since hypo patients are also frequently too low in the ranges for Vitamin D, B12 and ferritin, I suggest getting those tested as well.