Try taking a test light and checking to see that you have power at the source contact on the brake light switch which is located under your dash near your brake pedal. Then, press the pedal and see if there is power to the test light when it is connected to the wire which runs to the brake lights. Perhaps all you need to do is to adjust the brake light switch sensitivity which allows for movement of the button on your brake light switch. You are correct in that the signal lights and brake lights use the same element in a typical 1157 bulb. Good luck.
Check your fuses first.The brake light circuit is powered by a constant battery sourced fuse, as is the lighter and the horn. The turn signal circuit is powered by an ignition switched fuse, ie, the key must be on. The mechanical action within the turn signal switch interrupts the brake light circuit to the rear lamp on the side for which the turn signal is engaged.
