2010 Ranger will use 1 wire to each rear bulb for both turn signal and brake light signal.
The multi-function switch(turn signal stalk) is what controls that.
Wire from the brake pedal switch runs up to the multi-function switch, if turn signals are off then brake light power runs thru the switch and both rear bulbs light up.
If you turn on the turn signal then it will cut brake light power on that side and pass flashing power to the bulb on that side.
You could try a new flasher, there are two of these, one for turn signals, one for 4-ways
But your description that turning on the High Beams(using multi-function switch) causes a failure makes me think problem would be in that switch, not the flasher or wiring.