Just focusing on your thyroid related concerns, I see that your Free T4 is adequate, at 51% of the range, but your Free T3 is much too low, being only at 31% of the range. Many of us find that relief from hypo symptoms required Free T4 at the middle of its range, at minimum, and Free T3 in the upper third of its range, or as necessary to relieve symptoms. Scientific studies have shown that Free T3 levels correlate best with hypo symptoms, while Free T4 and TSH did not correlate.
Don't be concerned about the TSH. TSH is a pituitary hormone that is supposed to accurately reflect levels of the biologically active thyroid hormoes,; however, it cannot be shown to correlate well with either, much less correlate well with symptoms. After starting thyroid med, TSH testing ir useless. Also, since you have had a TT, testing for the antibodies of Hashi's is also unnecessary. You have no thyroid gland to be affected by the antibodies.
A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T4 and Free T3 as necessary to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not just test results. So you need to get your doctor to add some T3 to your meds to gradually increase your Free T3 level to relieve symptoms.
Other related areas of concern are the Vitamin D, at 37. It should be 55-60. B12 should be in the very upper end of its range. And your ferritin level is abysmal. It should be 70 minimum. So you need to supplement as necessary to raise these levels. Achieving those levels will mean a big improvement in how you feel, along with the benefit from getting your FT4 and FT3 optimized. .
Your FT3 is very low even though it is within the range so that can explain your hyposymptoms.
It also looks like your anemic but don't take my word for it, I'm definetly no expert. :) your ferritin level is way to low.
And yes, TSH can vary a lot from time to time.
I hope for you that your doctor listens to you as an individual and to your symptoms and treat you accordingly, probably raise your meds. And help you raise your ferretin.