In the words of an excellent 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."
So with the symptoms you have, and those low levels of Free T4 and Free T3, it is clear that you are hypothyroid. With your relatively low TSH, that is an indication of central hypothyroidism. With central there is a dysfunction in the hypothalamus/pituitary system resulting in TSH levels that do not adequately stimulate the thyroid gland to produce hormone. Doctors typically overlook the possibility of central hypothyroidism due to their over reliance on TSH.
TSH 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. A good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results and especially not TSH, which typically becomes suppressed when central hypothyroidism is treated adequately. I mention this just to alert you to a frequent problem: doctors who erroneously believe that a suppressed TSH automatically means hyperthyroidism, and want to reduce med dosage. In reality it is hyperthyroidism only when having hyper symptoms due to excessive levels of Free T4 and Free T3.
You can read about all of this in the following link. I highly recommend reading at least the first two pages, and further if you want to get into the discussion and scientific evidence of all that is recommended. You can also give a copy of the whole paper to your doctor and try to get understanding that you are hypothyroid and need thyroid medication. You should ask the doctor if he is willing to treat clinically, for symptoms, as described above. Also ask if he is willing to prescribe T3 type meds if needed. If either answer is no, then you will need to find a good thyroid doctor that will do both.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
I also suggest that you need some further tests. To assure there is no autoimmune system issue, I would test for Thyroid Peroxidase antibodies (TPO ab), and also since cortisol affects thyroid effects, I would test for morning cortisol. Since hypothyroid patients are so frequently deficient in Vitamin D, B12 and ferritin, those need to be tested and then supplemented as needed to optimize. D should be at least 50, B12 in the upper end of its range, and ferritin should be at least 70, and some sources say 100.
Please let us know how it goes when you discuss all this with your doctor. When you have any additional test results please post them along with reference ranges and we will be happy to help interpret and advise further.
Just because lab results fall somewhere within their range does not mean they are adequate for you; however, since lab results and accompanying reference ranges can very from one lab to another, we need to know the ranges shown on the lab report for those results. Also, please tell us about the specific symptoms you have.