You should not expect the Synthroid to be a fast fix. It takes weeks or months for the hormone to be gradually absorbed to an optimal level, and then it takes even more time for the damage done to the body to repair itself.
I don't know as much about low cortisol, except that it can make thyroid treatment more complicated. Low cortisol can make the body unable to process the Synthroid. You appear to have a smart and able doctor if he even thought to test your cortisol and to have found the Hashimoto's. Having a good doctor is more than half the battle.
Join us on the Thyroid board for support and information ~
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