TI's datasheet's says that the supply range should be between 4.5 to 36V. If that chip is connected at the regulator output, and the voltage regulator is unstable (wich i don't doubt), there will have a problem indeed. So something to try in meantime, even if i'm not a fan of it... replace the voltage regulator by a diode or two in serie.

Before i would prefer to monitor the voltage regulator output with a scope to see how bad it is...