With those test results and your description of the hypo/hyper symptoms, it appears that you have Hashimoto's Thyroiditis. With Hashi's the autoimmune system erroneously identifies the thyroid gland as foreign to the body and produces antibodies to attack and eventually destroy the gland. As this proceeds, the gland frequently forms nodules that leak hormone faster than normal. This causes hyper symptoms. When this slows then the patient returns to being hypo.
Both your Free T4 and Free T3 are too low in the range for many people. Free T4 is best at the middle of the range, at minimum. Free T3 needs to be high enough to relieve symptoms, typically in the upper third of the range, or higher.
A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not just test results and especially not TSH results. You need to find out if your doctor is willing to treat clinically as described. If not then you will need to find a good thyroid doctor that will do so.
In addition, hypo patients are frequently too low in the range for Vitamin D, B12 and ferritin. D needs to be about 55, B12 in the upper end of its range, and ferritin should be about 70. If not tested already, you should do so and then supplement as needed to optimize to those levels.
to boot i have a yo yo between hypo and hyperthyroid symptoms which is why i have been checked for lupus