You're not really going to get good low light action shots. Only way to eliminate blur from action shots is to use a faster shutter speed, which in turn, decreases the amount of light the sensor will be exposed to, which, unfortunately is also greatly needed, considering there is a lack of external light. So either the shot will come out blurry (use of slower shutter speed) or it will come out really dark to pitch black (use of faster shutter speed). You could, also increase the ISO, which will make the sensor more sensitive to light, allowing you to decrease the amount of time needed for the sensor to be exposed to light during the shot, but this in turn will greatly increase the chance of "grain" being present in the shot (Edit: Also, this is what is probably causing your "haze" effect on low light shots. It's probably due to the camera app aggressively increasing the ISO to adjust for the lack of light present).
Unfortunately, unless you use an app that will allow you to manually control the settings of the shot in a greater fashion than the G3 stock camera app, there isn't much you can do about how it adjusts those settings when taking a shot.