So I quit seeing her and got a referral to a new endocrinologist, with assurances that he was less rigid than my original doctor. He put me on Armor, which did seem to help normalize my body temperature and improved my skin. Armor is expensive and the tablets are huge and hard to swallow. My doctor also wanted me to take a half dose once a week, so now I'm taking Nature-Throid now, which is about 1/3 the cost, and it seems to be helping. I will know for sure when I get my tests in a few more weeks.
I wish I had asked for a referral to another endocrinologist instead of wasting time and money with the alternative practitioner. I thought I'd post this to help others going through the same thing.