D.W.
DO NOT QUIT THE MEDS for the sake of weight loss. Hashimotos and hypothyroidism are serious conditions. Here are the complications of having untreated hypothyroidism (going off your meds) and Hashimoto's
http://www.mayoclinic.com/health/hypothyroidism/DS00353/D...
http://www.mayoclinic.com/health/hashimotos-disease/DS005...
The hypothyroidism actually causes you to retain and even gain weight. The endocrine system is horribly complex, and there are several hormones that all affect your body's ability to gain/lose weight no matter what you do.
Synthroid has been on the market since the 1950's, so there's a wealth of information about its safety. Armour is a newer product that appears to have had some back-order issues recently, but appears to have been resolved.
Work with your endocrinologist, understand that the issues with your weight are completely out of your hands right now with your conditions, and make the best decisions you can with exercise/food choices. But, don't go off the medication. Hormones are all completely intertwined.
I know it's hard. Chemo made me gain a good amount of weight, and I'm afraid to lose some of it now because it was my only cancer symptom. But, I'd do ANYTHING to be healthy and make sure I do, even when I don't want to, because I've faced death in the form of cancer, and it sucks.