In the words of a good thyroid doctor, "The free T3 is not as helpful in untreated persons as the free T4 because in the light of a rather low FT4 the body will convert more T4 to T3 to maintain thyroid effect as well as is possible. So the person with a rather low FT4 and high-in-range FT3 may still be hypothyroid. However, if the FT4 is below 1.3 and the FT3 is also rather low, say below 3.4 (range 2 to 4.4 at LabCorp) then its likely that hypothyroidism is the cause of a person's symptoms."
Symptoms should be the most important consideration in diagnosing a potential hypothyroid patient. You have many symptoms of being hypothyroid. In addition the lab test results for free T4 and Free T3 fit the pattern of being hypothyroid, even though within range. The ranges are far too broad to be functional across their entire breadth, for everyone. The test results for T3 Uptake, Total T3 and T4, and Free T4 Index are outdated and not very useful.
From your symptoms, your Free T4 and Free T3 and TSH I'd say that you have central hypothyroidism. With central there is a dysfunction in the hypothalamus/pituitary system resulting in TSH levels that are too low to adequately stimulate output of thyroid hormone from the gland.
You are going to have to find a good thyroid doctor that will treat clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being constrained by resultant TSH levels. I say that because from treatment your TSH will typically become suppressed below range, which causes some doctors to erroneously diagnose hyperthyroidism. It is hyperthyroidism only if there are hyperthyroid symptoms due to excessive levels of Free T4 and Free T3.
Based on the outdated testing done by your doctor I am not optimistic that he will be a good thyroid doctor; however, you can get some useful info from the link in the following and give it to your doctor and see if he is willing to treat clinically as described above. If not you will need to find a doctor that will do so.
http://www.medhelp.org/user_journals/show/2019570/Diagnosing-Treating-Hypothyroidism-A-Patients-Perspective?personal_page_id=12021