Armour thyroid has both T3 and T4. Your blood tests show too much T3 and too low T4 while on Armour. Based on that, I would expect your doctor to switch you to a T4 only thyroid med. It makes sense that after switching to the new combo of T4 and T3 you had a thyroid storm experience - your T3 was already high and then you got even more of it in your system. Maybe it was supposed to be less T3 than the Armour? Even so, your TSH was so low that your thyroid was essentially shut off, yet your T3 was too high. I'd be curious to know what your blood results were without any T3.
In the short term, you might try only taking the L-Thyroxine or T4 component that your doctor prescribed and see how you feel.
A quick short term fix for too much T3 is to exercise. It will help use up the T3.
I think you could eventually get your thyroid sorted with supplements (iodine, selenium, vitamin A and B12, etc). If you aren't taking any thyroid hormone at all now, and feel ready to keep going with only supplements, there is a lot of info on the Iodine board. You can read my story in various posts I've written as well. I've gotten my TSH from a 9.4 to 3.7 in a few months with only supplements and diet, but it has been up and down, and I've had to constantly research symptoms to figure out what nutrients I was missing.
It sounds like you are already familiar with iodine, selenium and ATP cofactors. So you may already know this stuff, but just in case, here's what I learned over the past few months:
B12 - methylcobalmin sublingual cured me of brain fog and heart palpitations
Vitamin A - skip the betacarotene and go for only the vitamin A from fish oil. Start out with a huge dose - 50,000 IU and if you feel a difference, then keep doing that for a week and then drop to 10,000 IU a day. This cured me of night blindness and photophobia, and skin issues, and boosted my energy.
Iodine - build up slow! I got a big boost from Iodine and so I wanted more of that energy, but overdoing it too soon gave me a racing heart and insomnia. Start with 300-500mcg and see how you feel after a week. If you feel the same, keep going up. If you feel hyper or hypo, stay there for a few weeks before going up on it again. You may feel hyper, or hypo at times, but don't let that discourage you from taking it.
Don't take too much zinc. Zinc is great but when you get more than you need, your body creates metallothionein which prevents zinc absorption. The metallothionein actually binds to more copper than zinc and you can end up with "zinc induced copper deficiency" which will give you symptoms mimicking B12 deficiency: heart palpitations, fatigue, brain fog, etc. (Watch out for oysters which have a week's worth of zinc in them!)