In assessing lab results you always need to compare to the reference ranges. So please post ranges shown on the lab report for those test results.
Diagnosis for potential hypothyroidism should always be an integrated approach, including medical history, evaluation for signs/symptoms that occur more frequently with hypothyroidism, and also extended biochemical testing. You mentioned a family history with thyroiditis. Would that be Hashimoto's Thyroiditis?
Next, you reported several symptoms that are frequently related to hypothyroidism. And then your test results show high levels of autoimmune antibodies related to Hashimoto's thyroiditis. Your TSH is not yet above the high limit of its range, but then TSH is useful as an indicator of thyroid issues only when TSH is at extreme levels. Your Free T4 is mid-range, which is adequate. Your Free T3 of 2.9 is only about 35% of its range, which is not optimal for many people. Even though within range, the range is far too broad, and skewed to the low end, due to the erroneous assumptions used to establish the ranges for Free T4 and Free T3.
With you having the antibodies of Hashimoto's thyroiditis, there will be continual deterioration of your thyroid gland's ability to produce thyroid hormone. So you will need to take thyroid med to provide adequate levels of Free T4 and Free T3. The best description of what is needed comes from an excellent recent scientific paper, which states, "Hypothyroid symptom relief was associated with both a T4 dose giving TSH-suppression below the lower reference limit and FT3 elevated further into the upper half of its reference range. "
In addition, hypothyroidism is best defined as "insufficient T3 effect in tissue throughout the body due to inadequate supply of, or response to, thyroid hormone." the response is affected by a number of other variables. So you need to try and get tested for Reverse T3 (and Free T3 from the same blood draw), cortisol, Vitamin D, B12 and ferritin. From those tests you can determine if there are also other factors affecting you currently. I would even suggest that you delay starting on thyroid med until getting these tests done. Do you think you can get these additional tests done?
Yes, you have Hashimoto's thyroiditis. The sooner you start medication, the sooner you can start to feel better.
Red_Star is correct. You have Hashimoto's Thyroiditis and the result is that your Free T4 is only at 30% of its range, while your Free T3 is only at 26% of its range. Taken together it all indicates you are hypothyroid and in need of thyroid medication to raise your Free T4 enough to suppress TSH and raise your Free T3 level into the upper half of its range. In addition your ferritin is terribly low. It should be at least 100 so you need to either get a doctor to give you infusions of iron to raise your level faster, or you can supplement with a good form of iron like ferrous fumarate, ferrous sulfate, or ferrous bisglycinate, along with maybe 500 mg of Vitamin C to help with absorption and help prevent stomach distress from the iron. In addition you need the tests for Vitamin D and B12.
Note that when you start on thyroid medication it does not add to your prior level. The med causes TSH to drop and that reduces the output of natural thyroid hormone. Serum thyroid levels are the sum of both natural thyroid and thyorid med, so only when TSH is basically suppressed will further increases in thyroid med start to raise your Free T4 and Free T3 levels.