Its not uncommon here. A slugish thyroid can stress out the other hormone glands that keep you going. If I remember, a blood cortisol is just an average of the whole day. Now that you know you are low, you could try a 24 hr saliva adrenal test. It tells you your levels for four times during the 24 hour period. Mine was just low in the afternoon, not the other three times. It cost $150, a chiropractor ordered it for me.
If your adrenals are sluggish at all, they can be weakened by thyroid meds. It is very important to strenghten those adrenals and start out slow with your thyroid meds.....so you do not crash with exhaustion.