I re-read the OP ...
I would check the trailer wiring from front to back. It may be that someone has tied the left and right "brake/signal" lights together the same as the tail lights are. If this is the case, the brakes light WOULD work just fine. But, regardless of the signal light origin (left or right) both signal lights would blink every time. Then again, they would work that way if the headlights were on or off.
It still think it could be the ground. The ground may have just enough contact to [*** power for one type of light at a time. But, when you have all the lights on, all the power to light them all comes through the one ground wire. It may not have enough contact for the extra current drain and is "back-feeding" through the other lights. This is somewhat common when the ground is bad.