I’m not an expert, since I’m pretty new to this whole endocrinology thing but here’s a few things based on what I’ve learned so far.
The most sensible way to attack this problem is by discarding possible causes, one at a time. You are doing the Iodine thing so that’s one.
After you are finished with your iodine replenishment, take new labs and see where you’re at. If your thyroid function is still impaired, you can consider treating it with medication.
If things are not getting better, start looking at your HPA Axis (adrenal issues). Adrenal function and the HPTA axis are intimately related and issues in this department will definitely affect your sex hormones and sex drive, both directly and indirectly. Basic tests to see how you are doing in this department are: DHEA-S, Pregnenolone and Cortisol (preferably 4 sample saliva test to see how your levels behave during the day). If your adrenals are screwed up and very high, then you would discard serious conditions like adrenal or pituitary tumors. If they are low, you discard Addison’s. If the condition is serious, you can get it treated. If it is sub-clinincal, there are methods and supplements you can try.
After you get this fixed (if it needs fixing) and you still have symptoms, then you might want to consider TRT. Taking the previous steps before getting into TRT will improve the chance of it being successful. As KSman said, if you jump right into it you could start feeling better at first but then hit a wall if your other hormones are bottlenecking your progress.
I really hope this helps somewhat. Maybe someone more knowledgeable will chime in and correct this post if it needs correcting. In any case, I do believe that a systematic approach where you discard possible causes is critical. Eventually you will hit the nail in the head and you’ll get better.