I have Hashimoto's Thyroiditis, which is the most common cause of hypothyroidism. As you mentioned, finding a good doctor for this problem is not easy. Doctors are apparently hesitant to diagnose thyroid problems, and lots of people struggle to find someone who will take them seriously. Personally, I've been told by doctor after doctor that I just needed to count calories and get more exercise to feel better--as if I wasn't already. I was lucky enough to find a good doctor in my area recently, though, by doing some scouting around online. That might be a good start, if you haven't already.
I am also taking a more natural approach, and I am not on any type of medication at the moment. I am currently taking 1000mg per day of liquid vitamin D at my doctor's recommendation, as I had a deficiency that she thought may be causing most of my thyroid symptoms. I will have a follow up appointment soon, and we will do some tests to determine if I should see a specialist for some type of medication.
I have also been eating low carb at her recommendation, and have completely given up sugar. It is also highly recommended to give up gluten, but I haven't quite gotten there yet. All that being said, all of these things have really made a difference for me individually... maybe they will for her as well? Apart from looking online for a doctor who might be more open to helping than others, I also recommend that you take a look at hypothyroidmom.com -- there is some really great advice on her site, and it's been a big help for me.
Regarding the adrenal fatigue, I can't really offer any advice. I suspect that I may have this as well--it is commonly associated with hypothyroidism, and would explain my constant salt cravings--but I've not consulted with my doctor about it.
This is just the experience of one person, for what it's worth...