Sorry, it messed up my results and I don't know how to edit. It should be:
TSH 4.9 mU/L (0.40 – 4.00)
fT4 17 pmol/L (10-20)
fT3 5.3 pmol/L (2.8-6.8)
aThyro’gb 25 U/mL (<60)
aPeroxase 39 U/mL (35)
Iodine 132 ug/L (>99 is not Iodine deficient)
Due to the time difference it will take us longer than usual to receive and reply, so I will try to cover everything at once. First, I am not sure what those tests are: aThyro'gb and aPeroxase. Are those equivalent to those we usually see listed as TPO ab and TG ab? In other words tests for the antibodies of Hashimoto's Thyroiditis, which are Thyroid Peroxidase and Thyroglobulin antobodies.
Assuming that to be true, then your TPO ab test is indicative of the antibodies of Hashi's. That, and nodules that sometimes occur with Hashi's, might account for your swing in TSH from high to low and high again. With Hashi's it is quite common for nodules to form on the thyroid gland and leak hormone faster than normal, resulting in a cycle of hyperthyroidism and then back to hypo again.
That said, your current FT4 and FT3 level do not stand out as being very hypothyroid. Your FT4 is above mid-range, which should be adequate. Your FT3 is at 62% of its range, which is a bit light for some people, but not bad. Hypothyroid patients are frequently too low in the ranges for Vitamin D, B12 and ferritin. Low levels can cause symptoms that mimic hypothyroidism. Also, low levels of D and ferritin can adversely affect metabolism of thyroid hormone. Hypothyroidism is inadequate supply of or response to thyroid hormones at the cellular level (tissue). So it is not just due to the levels of FT4 and FT3. There are many other factors that can affect tissue thyroid effects. The most important measure of tissue thyroid effect is symptoms, of which you have a lot.
So if we look at some of the important variables for hypothyroid patients, your ferritin level is only half of what it should be for best effect. So it would be a good idea to supplement with about 50 mg of a good form of iron such as ferrous fumarate, ferrous bisglycinate, or ferrous sulfate. I would also suggest supplementing with B12 to optimize at the upper end of the range. I am not familiar with the meaning of the Holo TC test but your B12 is much too low. Besides those you really need to test for Vitamin D and supplement as needed to optimize at about 55. You should find a way to get that done.
Since cortisol is such an important part of the Endocrine system effects, especially with thyroid, I suggest that you also try to get tested for your cortisol levels, since your thyroid levels don't seem to account for your symptoms. Cortisol is an antagonist of thyroid hormone. It can be a problem is either too high or too low. The best test for cortisol is the diurnal saliva cortisol test done at 4 different times of day. Don't know if that is possible for you or not.
So my best suggestion for now is to get your ferritin and B12 levels optimized. Don't know why your doctor would think Vitamin D testing is unreliable. You really need to get that tested and optimize to a target of 55.
If you can get these done then
Very interesting that you would ask about DHEA-S and adrenal fatigue. I almost listed some additional tests for those, but decided that I probably should not overload you with tests to be done.
Over a long period of inadequate treatment for hypothyroidism, many patients develop cortisol insufficiency. That doesn't always show up clearly in test results, so symptoms must be assessed as well. The beset test for low cortisol is the diurnal saliva cortisol test taken at 4 different times of day. This measures "free" cortisol levels. The more commonly used test is morning serum cortisol, which is a measure of total cortisol, and not nearly as revealing. It is also important to consider symptoms. For that please click on your name and from your personal page, click on messages. I have sent a PM with info.