"How hard is it to change a turn signal switch?"

Not too hard, if you have a steering wheel puller and are comfortable doing things of this nature.

The bit about no power on the shared wire would have helped at the outset.

The power to the brake lights does in fact pass through the TSS. The switch opens the brake light circuit on the side to which you signal, substituting instead power supplied through the flasher. This is how one side is on solid and the other side blinks as you brake while signaling. It sounds like you have done a bit of diagnosis before posting, and that something is amiss in the area of the column. It could be a connector or broken wire after the switch.

I would recommend digging into the column (but not necessarily buying a new TSS yet) to continue your diagnosis.