More labs (as noted) are needed (Free T4, Free T3, Reverse T3, TPO & TgAb antibodies), which will help decipher between overt and sub-clinical. Even with extremely low T3 & T4 levels, TSH "usually" doesn't elevate that high. Not saying it can't or won't, just saying it's not everyday that patients with evident overt hypothyroidism present a TSH reading 6.00 < .
On your "High T" ... I see you are on HRT, so in essence your LH should be suppressed. my first paragraph was leaning towards this direction ...
TSH is one of several hormones produced by the anterior pituitary. LH & FSH (responsible for testicular activity) is also produced in this region, as is ACTH, prolactin, GH, and a few others. If you have a recent LH/FSH lab, it would be helpful? Again, those should be suppressed due to exogenous testosterone.
I'll keep an eye out for your labs and responses.