From your symptoms and your Free T3 result I'd say that you are hypothyroid. The most common cause for hypothyroidism is Hashimoto's Thyroiditis. Your TPO test did not confirm Hashi's, but to make sure you should also test for thyroglobulin antibodies. That test is TG ab.
I'm impressed that your doctor suggested Armour Thyroid, in recognition that your Free T3 is too low in the range. Free T3 largely regulates metabolism and many other body functions. Scientific studies have shown that Free T3 correlated best with hypo symptoms, while Free T4 and TSH did not correlate at all. You might want to ask the doctor to test for Free T4 instead of Total T4. If it shows the same relative to its range, and you do not have Hashi's, then you might also ask about testing for ferritin, and also selenium, to see if they are low in the range and possibly contributing to inadequate conversion of T4 to T3. Conversion issues are evidenced by the Free T3 level being significantly lower in its range, than Free T4.
Since hypo patients are also frequently too low in the range for Vitamin D, and B12, it would be good to test for those also. In my opinion I would try to get those done even before starting on thyroid med. If there is a good reason why your Free T3 is relatively lower than Free T4, then you might want to address that before starting on the med.
When you have additional test results, please post and members will be glad to help interpret and advise further. By the way, I would personally stick with your doctor, rather than assuming that an Endo is always a good thyroid doctor. Not the case. Many of them specialize in diabetes, not thyroid. Also many have the "Immaculate TSH Belief" and only pay attention to TSH. That doesn't work. Others use "Reference Range Endocrinology" and will tell you that a test that falls anywhere within the range is adequate.. Also very wrong.