Your Free T4/Free T3/TSH results are not "normal". They do fall within the so-called "normal" range, but that does not mean they are adequate. First thing to understand is that TSH is a pituitary hormone that is supposed to accurately reflect levels of the biologically active thyroid hormones, Free T4 and Free T3; however, TSH does not correlate well with either, much less correlate with symptoms. Further, each person has their own thyroid set points at which they feel normal. The range for TSH is based on data from large groups of people; therefore, trying to reliably identify if an individual patient's TSH is abnormal or not is very unlikely, unless it is extremely high due to Hashimoto's Thyroiditis. In addition the upper reference range limit for TSH has been pushed even higher than would be expected, just to avoid excessive numbers of false positive diagnoses. The result is excessive numbers of false negative diagnoses, like yours.
When a doctor does test beyond TSH the FT4 or FT3 are then compared to their reference ranges. Unfortunately the ranges are not based on healthy adults with no thyroid pathology. Instead the ranges are based on all the test data at a given lab, excluding only those with TSH that exceeds the range limit. So the TSH problem has even contaminated the ranges for FT4 and FT3. As a result, people with central hypothyroidism (like you) and even thyroid patients taking thyroid med would be included in calculating the reference ranges. Also with FT4 and FT3, there is a lot of inter- individual variability that also adds to the inability to identify abnormality for an individual being compared to group reference ranges based on group data. The end result of all this is that FT4 and FT3 in the lower half of the ranges should be cause for further evaluation, not cause to deny treatment.
Sorry to drag you through all that, but I wanted to provide the info , in case you wanted to understand the basic reasons why biochemical evaluation is inadequate. A good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free T3 without being constrained by resultant TSH levels. Symptom relief should be all important, not just lab results. You can get some good insight from this link written by a good thyroid doctor.
So with your symptoms and your Free T3/FT4/TSH levels I'd say you likely have central hypothyroidism. Central is a dysfunction in the hypothalamus/pituitary system resulting in TSH levels too low to adequately stimulate the thyroid gland.
Also, as mentioned hypo patient are frequently too low in the ranges for Vitamin D, B12 and ferritin, D should be 55-60, B12 in the upper end of its range and ferritin should be 70 minimum. So you need to get the D and B12 tested and supplement those and your ferritin as needed to optimize.
The most important question is how do you feel? Do you ahve any symptoms?
If you have no symptoms then you're fine.
If on the other hand you are having Hypo symptoms then no they are lower in the ranges than most people would generally feel well. So I would not be surprised that you have symptoms.
The rule of thumb is to have BOTH of the following. Understand that at the cellular level you ONLY use the Free T3 hormone. And to get T3, most of it is gotten by the body converting the storage Free T4 Hormone.
Most people need to feel well BOTH of the following:
1) Free T4 to be in the MIDDLE of the range if not a bit higher. This means 50% or a little more. Your test is ONLY 14% of the range which is well below the recommendation
AND - This means in addition to #1 above
2) Free T3 to be between the MIDDLE and the UPPER 1/3 of the range. This means between 50% and 67%. With many people needing it more towards the 67%. Your test is ONLY 17.7% of the range! Again woefully below even the lowest recommended rule of thumb level to feel well!
Your Ferritin is also extremely low. In order to metabolize thyroid you should ahve a Ferritin level of I believe around 70. You are testing at only 10.8.
That being said. You will likely have a hard time convincing a Dr you need medication. Because the modern Dr's who are scared of being sued are trained to look only at the numbers. And as long as the numbers are within the range, they are afraid to do anything. They could care less how you feel. You're TSH is what most Dr's believe to be "perfect".
So if you are symptomatic, you have to do a LOT of education and you have to be a STRONG advocate for yourself and you may very well have to go Dr shopping to find one who will listen to you and start you on medication.
Have you been tested for the Hasimoto's antibodies? This weold be TGab and TPOab. Hashimoto's is the #1 cause for low thyroid (Hypo) and is an autoimmune condition where your immune system incorrectly attacks the thyroid gland which results in the thyroid gland to procude less and less hormone. At some point you will need to get medication to provide enough thyroid hormone in order to feel well.
Also I would recommend that you be tested for Vitamin D3 and Vitamin B-12. As both of these if deficient can cause fatigue symptoms and it is not at all unusual for Hypo patients to also be low in one or both of these!