First TSH is a pituitary hormone that is affected by so many things that at best it is only an indicator, to be considered along with more important indicators such as symptoms and also levels of the biologically active thyroid hormones, Free T4 and Free T3. So yes, the TSH range is large because of the assumptions used to establish it, and also the intent to avoid false positive diagnoses by setting it higher than indicated by the data base. The only real value of TSH is to distinguish between overt hypothyroidism (TSH > 10), and central hypothyroidism, which is a dysfunction of the hypothalamus/pituitary system characterized by relatively low TSH with accompanying low levels of Free T4 and Free T3.
So, all that said, your TSH is high enough to at least trigger a test for TPO ab that was negative. At that point the doctor should have also tested for Thyroglobulin antibodies. Both tests are sometimes required to confirm Hashi's. So it is too soon to discount Hashi's and think about some non-thyroidal illness.
Your Free T4 is much too low in the range, consistent with having an elevated TSH. You should make sure they always test for Free T3 every time you go in for tests in the future. You need to know both. Free T3 is used by all the cells of the body and studies have shown that it correlates best with hypo symptoms. Based on the combined experiences of our members Free T4 should be mid-range at least, and Free T3 should be high enough in the upper part of its range to relieve symptoms. A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without regard for resultant TSH levels. Symptom relief should be all important, not just test results.
Prescribing 60 mg of Armour and then not even re-testing for 3 months is wrong. Serum thyroid levels are the sum of both natural thyroid hormone and thyroid med. As you start on the med, your TSH will go down in response and the output of natural thyroid hormone is diminished. The net effect is essentially little change in the biologically active thyroid hormone levels until the med dosage is increased enough that TSH is basically suppressed, at which point additional increases in med dosage will raise the Free T4 and Free T3.
T4 med has a half life of about one week, which means that it takes about 4 weeks to reach over 90% of the final effect on serum levels. T3 has a half life of less than a day, so it reaches full effect in a week. I suggest that you should call your doctor and relate all the symptoms you are having and ask to be re-tested. This time you should insist on Free T4, Free T3, TG ab, Reverse T3, Vitamin D, B12 and ferritin. Cortisol is also important, but you have done one of those tests. By the way was that the 24 hour saliva cortisol test?
I am somewhat hopeful that your doctor will be willing to consider clinical treatment as described above, since she at least was willing to prescribe Armour Thyroid, which is a good sign. When you have more test results, please post, along with reference ranges and we will be happy to help interpret and advise further.