If your tests are going to be within a couple of months, I agree, wait to take iodine till after them. I think it can take at least a couple of months for TSH levels to settle down, and maybe as many as six months. In the meantime, I don't see any reason not to start selenomethionine 200 mcg, vit C 2-4 grams, magnesium 400-600 mg, and a good B complex, if you're not already on them. All these will help your general health, and prepare you for taking the iodine as safely as possible. I understand with Hashi's you would have the best chance of doing alright with iodine if you take the selenium for awhile first. And if you do show elevated antibodies, consider dropping wheat and all gluten from your diet to reduce your levels of inflammation.
Especially if your adrenals are weak, consider starting each morning with 1/2 - 1 teaspoon good salt, and drinking at least half your body weight in ounces of water per day. And be sure your ferritin levels are a minimum of 70-90.
The recommendation for the current formulation of Armour is to chew it before swallowing. This will break up the cellulose binders and help you maximize your dose. I haven't had to do this, so I don't know what it's like! Ideally it would be nice if you could take a sublingual version that would disperse in your mouth, such as Erfa Thyroid from Canada or the new generic thyroid from Acella. Regardless, I don't know why Iodoral would interfere with thyroid extract. After all, iodine makes up a great part thyroid extract. T4, for instance, is 4 atoms of iodine attached to one atom of tyrosine.
However, take your thyroid separately from calcium, iron, and estrogen, as they will bind some of the thyroid hormone while mixing in the stomach.
Hope this helps. Welcome! You've already done a lot of good, careful work