Before commenting further, please post the reference ranges shown on the lab report for those test results.
I feel like I am the same as you. I have the exact same type of symptoms. Although I am gaining weight for no reason that I can see. It is not really debilitating but certainly is impacting me.
TSH 2.23 (0.20-4.00) mU/L
Thyroxine Free 14.6 (9.0-23) pmol/L
FreeT3 5.0 (3.5-6.5) pmol/L
Help?
Well at least that is a start; however, be aware that 25 mcg is not going to do anything for you. It is only a small, starter dose. In item 10 on page 13 of the link I gave you it explains why you should expect no significant change from that dose. " ....serum thyroid hormone levels are a sum of both natural thyroid hormone and thyroid medication. As medication dosages are increased the production of both TSH and natural thyroid hormone is diminished. As a result, equilibrium serum levels are not increased with small starting doses of thyroid medication. Only when TSH is no longer stimulating natural thyroid hormone production, or is suppressed, will serum thyroid levels reflect further increases in thyroid medication."
So you will have to continue to push the doctor to re-test every 4-5 weeks and raise your dosage enough to raise your FT4 and FT3 levels adequate to relieve symptoms.
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It is not at all unusual for Free T4 and Free T3 to remain the same or even drop when starting on thyroid medication. The reason is that the medication will cause a reduction in TSH, which then causes a reduction in output from the thyroid gland. Since serum thyroid hormone is the sum of both natural thyroid plus thyroid med, the net effect is minimal. Only when the dosage is enough that TSH is suppressed enough to no longer stimulate natural thyroid hormone production will serum thyroid levels reflect further increases in thyroid medication. You can read about this in Rec. 13 on page 13 of the following link.
http://www.thyroiduk.org/tuk/TUK_PDFs/The%20Diagnosis%20and%20Treatment%20of%20Hypothyroidism%20%20August%202017%20%20Update.pdf
In your latest test results your Free T4 is too low, at only 27% of its range. It needs to be at least mid-range. Your Free T3 is way too low, being at only 10% of its range. Free T3 is best in the upper third of its range, and then adjusted as needed to relieve hypo symptoms. The FT3 being lower in its range than FT4 is also an indication of inadequate conversion of T4 to T3.
I think you are going to run into several obstacles to getting the treatment you need. First is that you may have difficult getting the doctor to continue to raise your thyroid med, especially if your TSH becomes suppressed and is interpreted as meaning you have become hyper. That is the case only if you have hyper symptoms due to excessive Free T4 and Free T3. You can also read about this in the link above. Second is that you may have difficulty in finding a good thyroid doctor that will treat clinically by 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. Third is that you are likely to find it difficult to get a doctor to prescribe T3 type meds like Armour Thyroid, NatureThroid, or Cytomel.
If you find that you need a good thyroid doctor that will treat clinically, for symptoms, and also is willing to prescribe T3 type meds, then tell us your location, and perhaps we can suggest a doctor in your area that has been recommended by other thyroid patients.
Sometime in the near future you need to re-test the Vitamin D and B12. Also a good idea to test for cortisol if possible. Both D and cortisol has an effect on metabolization of thyroid hormone.