It will have a hot feed and the fact that it's hot on both sides all the time is because like you said it's bad. The next thing in the circuit is the turn signal switch. Try slowly moving the turn signal lever a little bit each way without going far enough to make the signals work and see if the brake lights come on. The way it works is the power from the brake light switch goes to the turn signal switch and the is split through a circuit to each rear light. Just because the turn signals work doesn't mean the switch is good.