If I'm reading this right, definitely sounds like it's overloaded:
For just the two new lights:
2 x 100W = 200W
P = VA (power = volts * amps)
or
A = P/V
Voltage is ~12V (well, in a perfect system...)
200W / 12V = 16A
As per a few sites I've read, depending on the insulation on the cable 14AWG can handle 15 - 20 amps (I even read 25 amps, for the high end).
So you're slightly over the limit on the lower end of the scale, with just the two new lights. Based on the fact you say it works fine with just the driving lights, I'm guessing your gauge wire is fine for that (so it can probably safely handle up to 20amps give or take). Add in your brights though, and you're pulling too much current through the wire.