"In Texas the car in the intersection has the right of way, even if they crossed in front of you."
You are sorely mistaken on this point. The car turning left must first yield to oncoming traffic. Where do you come up with this stuff??
From the Texas code :
Sec. 545.152. VEHICLE TURNING LEFT. To turn left at an intersection or into an alley or private road or driveway, an operator shall yield the right-of-way to a vehicle that is approaching from the opposite direction and that is in the intersection or in such proximity to the intersection as to be an immediate hazard.
With that kind of driving knowledge, I'm amazed that you don't get T-boned on a regular basis.
To the OP, about the only thing that could cause you to be even partially at fault were if you were travelling at an excessive rate of speed. If no one is saying that, I don't see how they could say that you were at fault.
If Geico doesn't agree, your only recourse will be to file suit against the other driver. Geico will be obligated to defend the other driver and explain to a judge how they came to the conclusion that you were at fault.