Liver profile looks good, so thats not a factor that I can see.
The more SHBG you have, the more SHBG+T you have and that can give higher TT levels.
If you are saying that others in your family have the same lower body temperatures, there are two possibilities. One is that the thermometer is not good, the other is that others in the family might have the same iodine deficiencies. Do they sometimes feel cold easily, or is the climate warm that that would never happen?
You can try to lower E2 and SHBG might follow.
These really did not look good:In another report 3 months back :
Total T3 :110 ng/dl (60-200)
Total T4 : 6.95 ug/dl (4.5-12.0)
Total TSH: 2.49 UIU/ml(0.27-5.50)
Low iodine can do that. Did you read about iodine replenishment [IR] in the thyroid basics sticky?
So I see two issues, SHBG and thyroid. You would be better off if the body temps and thyroid levels were from low iodine, as that is correctable. Note that I strongly recommend that one have an identifiable source of selenium, especially during IR.