What were the actual results and reference ranges shown on the lab report for the thyroid related tests?
A reaction to both types of med make me think it is not likely the med itself. There are reports of reactions to thyroid med being caused by low ferritin (iron related) or low/high cortisol. So in addition to the tests recommended by Emily I suggest testing for ferritin as in indication of iron status, and also cortisol. The best cortisol test is the diurnal saliva (free) cortisol panel of 4 tests. Doctors predominantly do not order this test, but instead order a morning serum (total) cortisol test, which is more difficult to interpret, but at least may give an indication of whether cortisol is involved.
In general, how would you describe your day. Do you awake refreshed and gradually go down hill as the day progresses? Or do you have trouble waking up and don't ever feel refreshed from sleeping, and then your energy level tends to improve as the day progresses?
Have your doctor check your Vitamin D, B12 and iron levels