You seem to have totally usless doctors and that is a huge part of your issue. You cannot have a reasonable chance to determine a cause, and a fix, if you cannot get proper and sufficient blood tests.
I also wonder if there is more than cortisol and thyroid going on here. I would suggest you get your sex hormones tested and be absolutely sure to get testosterone tested. yes believe it or not, women have testosterone and NEED it in the proper levels. In fact, normal women in prime of life have 300% more testosterone by volume than estrogen!
Contrary to common lore, testosterone actually is a calming hormone. Steroids are a different matter and the "roid rage" is associated with adrenal hormones not testosterone in NORMAL and sane levels.
What other symptoms do you have? Do you have menstrual issues? If so what kind. Thyroid can play a roll with irregular periods etc. But so do the sex hormones (Estrogen - specifically Estradiol, progesterone, and testosterone.) if you get tested also make sure you get tested for SHBG and DHEA.
Elevated cholesterol are also signs or symptoms of low thyroid but also low testosterone. How are your cholesterol levels?
Hypothyroid patients are frequently deficient in Vitamin D, B12 and ferritin. Ferritin is a storage form of iron that is a precursor to serum iron levels. Ferritin is very important for good hair growth. Women are much more prone to be low in iron than women. I don't see how the doctor can refuse you if you tell him all this. In addition, you need to know your Free T4 and Free T3 levels, and even Reverse T3 as well. There is a really excellent recent scientific study that concluded that, "Hypothyroid symptom relief was associated with both a T4 dose giving TSH-suppression below the lower reference limit and FT3 elevated further into the upper half of its reference range." Have you even had a Free T3 test done before?
As for cortisol, if low there is certainly something that can be done by taking cortisol med. If it is high taking a supplement called Rhodiola has been shown effective in reducing cortisol.
Along with testing for the thyroid hormones, in view of your reaction to thyroid med you definitely need to be tested for cortisol and ferritin to see what needs to be corrected.
No way you should just accept your condition. Are you aware of the negative reactions that can occur when increasing thyroid med, due to low or high cortisol or low ferritin? That possibility makes the most sense to me. So for that reason you absolutely need to be tested for Free T4, Free T3, Reverse T3, cortisol, a full iron panel (serum iron, TIBC, % saturation, and ferritin). I think if you get those done you will see what is causing your issues and what needs to be done. Hypothyroidism is not just about serum thyroid hormone levels. Hypothyroidism is insufficient T3 Effect in tissue throughout the body, due to inadequate supply of, or response to, thyroid hormone. So you see it is not just the supply of thyroid hormone, but the effect of other variables, on Tissue T3 Effect.
It is vital that you get those tests done. Is there a way to make it happen?
Also, to give you assurance of the importance of these tests, I am sending you a PM with a link to info on the effect of low or high cortisol and low iron.
You mentioned previously that you wanted to , " check for the thyroid antibodies, reverse T3, cortisol, all vitamins just in case, ferritin storage of iron, everything. " Which tests have you gotten done and what were the results and reference ranges shown on the lab report?