My personal opinion (and I have a thyroid gland) is that those who have had a thyroidectomy should be given a combination dose of T4/T3. T4 is levothyroxine a synthetic thyroid hormone which should convert to T3 (liothyronine) the only active thyroid hormone and it is need in our billions of receptor cells and it is T3 which drives our whole metabolism.
The British Thyroid Association have removed T3 (liothyronine) from being prescribed and they had a perfect excuse in that the supplier tripled (or more) the cost of it. You can source it yourself but no info is posted on the open forum but through a private message/.
Adding some T3 to a reduce T4 may be very helpful as Research has shown that a combination works for many who cannot get well on T4 alone.
The very first thing you must do is get a new blood test. It has to be the very earliest possible and fasting (you can drink water). Also allow a gap of 24 hours between your last dose of levo and the test and take afterwards.
Thyroid hormones should be taken first thing on an empty stomach and wait about an hour before eating. Food interferes with the uptake.
Ask GP for TSH, T3, T4, Free T4 and Free T3, B12, Vit D, iron, ferritin and folate. We can be deficient in minerals/vitamins and they have to be optimum.
Always get a print-out of your results with the ranges (some labs wont do the 'extras' if TSH is in range). The most important FT4 and FT3 can be done privately through one of our labs. I shall give you a link and you'll see why they are important.