I'm a big proponant of testing for cortisol and it's good to get the test. But did you ever get your free T3, free T4, and antibodies tested? Your TSH and symptoms indicate hypo and even if your docs disagree, it looks like you really need the proper testing and perhaps treatment. And if your cortisol levels come back low you still will need a good doc to treat this too.
Keep looking for a doc who knows something about thyroid and adrenal treatment. They are out there. Try posting here for a recommendation and ask local pharmacists if they have any docs precribing natural thyroid hormones-- not a guarantee, but at least indicates someone more open to that treatment.