With those symptoms and your relatively low FT4 and even lower FT3 level, it is apparent that the Hashi's is adversely affecting the ability of your thyroid gland to produce enough hormone for your needs. So you need to get started on thyroid medication. a good thyroid doctor will treat a hypo patient clinically by testing and adjusting FT4 and fT3 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 results when taking thyroid med. Many of us have found we needed FT4 about mid-range, and Free T3 in the upper third of its range, or as needed to relieve hypo symptoms. If you think your doctor might be resistant to clinical treatment, then I highly recommend reading at least the first two pages of the following link, and more if you want to get into the discussion and scientific evidence for all that is recommended. Then you can use the info with your doctor, to get the treatment needed.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Also, since hypo patients are so frequently deficient in Vitamin D, B12 and ferritin, I recommend testing those and then supplementing as needed to optimize. D should be at least 50 ng/mL, B12 in the upper end of its range, and ferritin should be at least 100.
Do you think you will be able to get your doctor to treat clinically, and do the additional testing?
Your test results show Hashimoto's Thyroiditis. With Hashi's the autoimmune system erroneously determines the thyroid gland to be foreign to the body and produces antibodies to attack and eventually destroy the gland. As this proceeds, the output of thyroid hormone from the gland is gradually diminished, and the pituitary output of TSH increases in a vain attempt to maintain adequate production of thyroid hormone.
Just because your Free T4 and Free T3 results are still within their reference ranges does not mean they are adequate for you. The ranges are far too broad to be functional for everyone, plus each person can have different requirements for FT4/FT3, in order to feel their best. Your Free T4 level is only at 35% of its range, which is lower than many of us have found to be required. Your Free T3 is even lower in its range, at 19%. Many of us have found that we needed Free T3 to be in the upper third of its range, and adjusted from there as needed to relieve hypo symptoms.
Even more important than those test results are symptoms. Treatment decisions for you should be primarily based on symptoms, rather than just test results. So please tell us about any symptoms you have.