You could also be having a reaction to fillers/binders in the medications.
You might ask your doctor about trying Tirosint, which is gel cap and has no fillers/binders, so is hypoallergenic.
I agree that you need to have Free T3 and Free T4 tested, along with adrenal function. Unfortunately, just an AM cortisol is not sufficient.
I hope you take this seriously, but one of the complications of treating hyporthyroidism is if a person also has hypoadrenalism. When both are low they kind of balance each other. But when you start treating with synthroid, the adrenals can't keep up with the increased metabolism.
Your endocrinologist should test your AM cortisol, if it's low or 'low normal' you need to rule out additions disease. (It's not uncommon that a person with one autoimmune endocrine disease has another)
Also should check your testosterone levels as well as the AM Cortisol.
Could be something else. Do have these checked if they haven't.
Please post your blood lab results along with reference ranges.
TSH is a screening tool at best and COMPLETELY inadequate to adjust medication. In fact it is just short of worthless once you start medication.
You MUST demand if you don't already have them to be tested for the two thyroid hormones your body actually uses. These are FREE T4 and FREE T3. The "free" tests are different than "total". As the free tests only measures the amount of hormone that is isolated and "free" from being attached to a protein molecule. Once attached the hormone becomes biologically inert or useless.
A Dr who is prescribing a medication with a T3 component in it such as Armour borders on malpractice in my opinion if they are not testing you for Free T3.
What other symptoms do you have other than the edema?