The symptoms you mentioned are frequently related to hypothyroidism, and your Thyroid Peroxidase antibodies test is indicative of Hashimoto's Thyroiditis, which is consistent with your high TSH that is above range. So yes, you do need to be started on thyroid medication. You also need further testing. Specifically you should be tested for the biologically active thyroid hormones, Free T4 and Free T3 every time you go in for tests. If they are in the lower half of their ranges, then that too is consistent with hypothyroidism. Hypo patients are also frequently deficient in Vitamin D, B12 and ferritin, so those should also be tested and then supplemented 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.
The biggest problem you face right now is finding a good thyroid doctor that will treat clinically, by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms without being influenced by resultant TSH levels. Most doctors only pay attention to TSH and will medicate only enough to return TSH within range. That doesn't work for most people. There is a very important scientific study that recently showed that hypothyroid symptom relief was associated with both a T4 dose giving TSH-suppression below the lower reference limit and Free T3 elevated further into the upper half of its reference range. So that should be the ultimate goal, along with optimizing Vitamin D, B12 and ferritin. If your doctor is unwilling to treat clinically as described, then you will have to find one that will do so.