I don't have a G2, but this should be the same for pretty much every camera, cell phone or otherwise. Are you using the flash? Have you tried adjusting the ISO value in settings?
In any camera, the sensor needs a certain amount of light to properly expose the image. The primary control for that is the shutter speed. In bight light, a very brief exposure can be used, which freezes motion. In dim light, the shutter speed needs to be much longer, to allow enough light to hit the sensor. The longer the shutter is open, the more motion blur you'll have. Changing phones won't eliminate this problem.
There are two things you can do. One, increase the amount of light. Either turn on more room lights, or use the flash. Two, increase the ISO speed in settings. Increasing ISO essentially increases the brightness of an image after it's captured, allowing the camera to use a shorter shutter speed. Think of it kind of like music: if you have a song that is very quiet, you can turn up the volume (increase the ISO) to make it louder (brighter). But when you do that with music, you'll also hear more hiss. Same with a photo: turning up the ISO will result in more "noise" in the image, which shows up as a grainy look.
Something else to check. If the G2 camera app has any sort of scene modes (e.g., HDR, Portrait, Nature....) see if there's one called Low Light or something like that. It might, or might not, help with your problem.