You are in a weird spot. I see how you would think that management should cover some of the cost of the utilities. However, looking at it from the view point– You probably could have done something to stop the heater/ air from running non-stop, even if that meant flipping a breaker to shut the unit off. Although inconvienant, it would prevent you from having a huge utility bill. I’m not sure about Texas, but November, December, January and even Feburary have historically been higher opporating cost for utilities– simply because its the coldest months. So it would be expected that the heating cost would go up quite a bit, even as much as double.
Its clear there was a mal-function with the heating / airconditioning unit. But still, it is not necessarily the responsibility of the landlord– unless you can prove that they were extremely neglegant at repairing the unit.

As a tenant we have what we call Renter’s Right to Minor Repairs. Your landlord needs to keep the structure of the building in sound that’s including the electricity. Now if you feel like there’s something wrong with the billing, you can call up one electrician and check on it. If you found out there’s something your landlord is not doing right with the billing talk to him and inform him you your rights about this matter.