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Can you pinpoint anything that happened in 2011 when you started feeling this way?
Your testosterone was low, but you started TRT already. I am looking forward to seeing the new number. Many guys have to use T creams with 200 mg/mL (200 mg per day) to get TT over 700 ng/dL, so keep that in mind. How do you feel now after a while on TRT?
I am assuming that you got a sleep apnea study, right?
Do you suffer from allergies (sinus, etc)?
Your TSH is high. There may be some sub-clinical hypothyrodism. Your symptoms also point to that direction. Chris can add more info on his thoughts on your thyroid values.
Your salivary cortisol does not look bad (a little higher than normal in the evening).
I do not trust salivary hormone levels (except cortisol) for any diagnosis, so won't comment on those.
I would love to see your iron and ferritin blood levels. Your high TSH and symptoms may indicate that there may be an issue related to mild anemia.
If everything above does not lead to answers, I would get tested for Lyme disease and also have my immunoglobulins checked.