I too also have this question as I am in a similar situation. My cortisol was barely within range (2.2, 2-18), but I've had sleeping problems and puffy eyes far before my abnormal thyroid results came in and now I'm most def hypo too. Couldn't a ACTH stimulation test be done to get the measure of your adrenal/cortisol?
I think you need more checked and an ultrasound of your adrenals to rule our that your hypothyroidism is primary and not secondary. Adrenal issues can mess with your thyroid too.
I don't usually post on this forum, but per your request, I've slipped in.
As you know a single cortisol is not sufficient to diagnose an adrenal issue; you would have to have the 24 hr saliva test, though many doctors don't recognize it as a valid test. We've had many members in the thyroid community tested for both adrenal insufficiency and hypothyroidism.
It's said that adrenal issues must be treated prior to treating a thyroid condition, but it must be noted that when the thyroid fails, the adrenals kick in to try to take up the slack, so when thyroid treatment is initiated, it often takes time for everything to return to normal. That's one reason why many people feel worse once they start on thyroid medication. The adrenals don't always need special treatment; it's just that the body has to re-balance itself once thyroid hormones are available. That's one reason why it's best to start out thyroid treatment at very low doses and work up slowly - to let the body balance out slowly.