I am betting it is a ground issue
not sure what this means , "wired through my existing wires."
Anyway, the brake signal and turn signal uses the same wire and the same filament in the bulb.
You don't mention the tow vehicle.
Does it have separate turn and brake lights?
If so, you may need a converter on the vehicle.
Also, a bad, or light duty flasher unit on the vehicle, can cause this.
Try a new heavy duty flasher.
Brake light bulbs are two filiments. Do you have the bulbs in right ?
sounds like you think it is a bad ground in the trailer? Something common to both sides. I am going to go do some more tests.
http://www.harborfreight.com/four-way-trailer-light-tester-66526.html
$5. I have a multimeter so I use it, but if you don't have a meter, these are hard to beat.