In diagnosing possible hypothyroidism, the most important consideration is symptoms. You have a number of symptoms that are indicative of being hypothyroid. Even though your TSH is well within range may only indicate central hypothyroidism. With central, the hypothalamus/pituitary system is dysfunctional and does not produce enough TSH to adequately stimulate the thyroid gland to produce thyroid hormone. Even though your Free T4 and Free T3 are within their so-called "normal" ranges does not mean that all is well. Due to the erroneous assumptions used to establish the ranges, results for FT4 and FT3 in the lower half of the range is reason to suspect hypothyroidism. Note the following quote from 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."
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 constrained by resultant TSH results. Symptom relief should be all important, not just lab test results.
So you are going to need to find out if your doctor is willing to treat clinically as described. If not, then you will need to find a doctor that will do so.
As a side note, hypo patients are frequently too low in the ranges for Vitamin D, B12 and ferritin. D should be about 55 min. B12 in the upper end of its range, and ferritin should be about 70 minimum. So be sure to get those tested and supplement as needed to optimize.