There should not be power on the brake output wire unless the brake controller is sending back power. It sounds like something might be miswired at the 7-Way. Take a look at the picture I have included that shows the pin out of a vehicle side 7-Way trailer connector to make sure yours is wired the same way.
If it isn't then put the wires in the appropriate places. Also check to see if any of the wires are touching each other inside the 7-Way. And if the 7-Way is dirty or corroded that... view full answer...