Taking thyroid med causes TSH to go down, yet FT4 and FT3 do not increase significantly until TSH is suppressed to the point that there is no natural thyroid being procued by the gland. I would not worry about the TSH level. Just get your FT4 and FT3 levesl high enough to relieve hypo symptoms, along with necessary Vitamin D, B12 and ferritin.
Hashimoto's Thyroiditis is usually characterized by high TSH. Why do you think you have Hashis?
Total T4 and Total T3 tests represent the total serum levels. Most of that is bound to protein and is thus inactive. The small portions free of protein, are called Free T4 and Free T3. Those are the more important tests. Both of your tests appear to be for Totals, not the Free portions. So you really need to be test4ed for Free T4 and Free T3.
Even more important are symptoms. From the following list of symptoms typical of hypothyroidism which do you have.?
Tiredness.
More sensitivity to cold.
Constipation. (have to use laxative or fiber)
Dry skin. (have to use skin creme)
Weight gain.
Puffy face.
Hoarse voice.
Coarse hair and skin.
Muscle weakness.
Muscle aches, tenderness and stiffness.
Menstrual cycles that are heavier than usual or irregular.
Thinning hair.
Slowed heart rate, also called bradycardia.
Depression.
Memory problems.