Yes, your TG ab test result shown as high indicates Hashi's.
In assessing a patient for possibility of hypothyroidism, the most important is an evaluation for symptoms that occur more frequently with hypothyroidism than otherwise. You have many of those. Next should be evaluation of thyroid related test results to support symptoms evaluation. Your Free T4 of 1.1 is only at 25% of its range. Your Free T3 of 3.2 is at 50% of its range. Many doctors would tell you that since those are in range, you don't have a thyroid problem. That is incorrect because the ranges are far too broad due to the erroneous assumptions used to establish them. The ranges are also skewed to the low end. Also, every person has different thyroid parameters at which they feel normal. Note the following 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."
In view of all this, it was good to hear that your doctor wants you to start on thyroid med, namely Armour Thyroid. That would be a good thing for you. Note, however, that a starting dose of Armour will not be additive to your current levels. That is because the med will cause TSH to drop and thus your output of natural thyroid will also drop. Since your serum levels are the sum of both natural thyroid hormone and thyroid med, the net effect will be essentially no change. Only when the med dosage suppresses TSH will further increases in thyroid med start to raise your Free T4 and Free T3 levels. What you will need will be enough increases to get to a dosage that results in symptom relief per the following, as concluded from a recent, scientific paper. "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 correctly defined as "insufficient T3 effect in tissue throughout the body due to inadequate supply of, or response to, thyroid hormone". So it is not only related to thyroid hormone but also the response. The response can be affected by variables such as Vitamin D, ferritin and B12. So you need to be tested for those and then supplement as needed to optimize. D should be at least 50 ng/mL, ferritin should be at least 100, and b12 at the upper end of its range. Cortisol is also important, so be sure to get that tested since either low or high cortisol has an effect.
If you want to confirm what I say and also have a paper to review with your doctor, then click on my name and scroll down to my Journal and read at least the one page overview of a full paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective.