Let me just start by saying that if you go to the websites of either manufacturer of Armour (Forrest Pharmaceuticals) or Synthroid (Abbott labs) or look up the name of the medication on the Internet along with the word "contraindications" it will mention Adrenal Insufficiency in the 'contraindications'. Ah...and the reason I know that is that I've been where you are....
I became progressively ill while on thyroid meds, mine however was VERY dramatic, all within a few months. I had a TSH of just outside normal and that was the ONLY thing my Dr. used as a diagnostic tool...big mistake. I was advised on the phone by Abbott Labs, who makes Synthoid to discontinue my meds as the only time they saw this reaction was in patients with Cushings and those with undiagnosed Adrenal Insufficiency. Thyroid meds can and do overdrive the adrenals, which in the beginning manifests with too high of cortisol then lowers as the years go by. This only happens when the adrenals are too fatigued to keep up, yet they will overdrive like a 'whipped horse' until they just can't keep up.
I'm not in any way saying to discontinue your meds. You obviously have a real thyroid problem...but the long and short of it is this...IF you have compromised adrenals even if you aren't taking thyroid meds, they will overdrive themselves to keep up with demand, then they will slowly lose vitality. The question is WHY are they overdriving? Which is what you're trying to figure out.
Are you familiar with Hans Selye and his well known theory of Adrenal stages? "Shock, Countershock, Resistance and Exhaustion". Check out http://www.chronicfatigue.org/History.html
Start with that. My 2 favorite books are Dr. James Wilson's ADRENAL FATIGUE and Dr. Poesnecker's CHRONIC FATIGUE UNMASKED 2000. (It's really about adrenal problems, don't let the title fool you.)
There is no harm in seeing the Neurologist, or having an MRI of your Adrenals, Pituitary (a tumor can raise cortisol dramatically), Hypothalamus etc. if your Dr. wants as it is sometimes prudent to rule out the obvious, especially if insurance will cover it.
But, if you are in the beginnings of the Alarm Phase you're at the PERFECT time to begin diet, lifestyle, supplement changes etc. which could keep your adrenals from burning out. ALL the symptoms you mention CAN be adrenal related, but can also mask other problems.
TRUST YOUR GUT INSTINCT! If you're the praying type (like I am) I love the verse, "Does any one lack wisdom? Let him ask of God who gives generously to all, without finding fault". I spent SO much time listening to Dr.s (even "good" endocrinologists, not one in 100 knows anything useful about adrenal problems). If you are CERTAIN at some point that it's adrenal related there are some good alternative clinics in the US that deal specifically with these problems, some that will consult by phone. Although usually it's all out of pocket, not covered by insurance.
P.S. If your progesterone is low, try natural Life-Flo Progesterone Cream. It REALLY helps. The cheapest place I've been able to find it is http://www.swansonvitamins.com.
But make sure you DON'T use it within 5 days of a Cortisol test as it will mess up the results.