The dose you mention, 12.5 mcg, is a very tiny dose. A teaser dose. When we take T4 meds, the body in a number of ways tries to compensate, either by reducing its own T4 production, or by decreasing conversion of the inactive T4 to the more active, potent T3.
T3 is very important for keeping the brain up to speed. I'm wondering whether you might have experienced a "net less" T3 as a result of the teaser dose. That might explain the brain fog.
How were you originally dxd? Did you have hypoT symptoms before you tried the Synthroid? If you did not receive hard copies of any labs that were done, I would encourage you to get those copies. See what tests were done, and what the numbers were.