For a budget of $300, you won't get a lot better from the new camera compared to your old one.
Tripod is your best friend in low light. A gorilla pod is also quite useful if you don't want to lug around a tripod.
For low light still photos, use low ISO and slow shutter speed. If you are taking a photo of someone standing in the foreground, then use flash for the person in the foreground and a slow shutter speed for the background.
For low light action photos, there is no $300 camera that can capture noise free sharp and clear photos. Instead use flash and time your shot to freeze the action. Or you can use your camera's video function. For indoor video, a tripod can also improve the video quality.
For concerts (if your kids are the performers), get as close as you can and use flash if allowed. If flash is not allowed, then use a tripod and video (try not to zoom in if possible).
Try not to zoom during video! Zooming during video is what amateurs do a lot. Professionals do zooming very selectively. Learn to move your camera around to shoot from different angles and different distance (use your legs). Zooming will degrade image quality, even if you use optical zoom. Zooming will decrease the amount of light reaching the sensor and will increase noise and degrade image quality. Zooming during video may also throw the camera out of focus, ending up with a blur. Furthermore, it is far more interesting to take shorter videos from different angle and different distance (use your legs to change your position) of the same subject, giving the viewer different perspectives, than to stand in the same spot zooming in and out. Zooming in and out too much may distract the viewer and make them feel dizzy. Next time when you watch a movie, pay attention to how the professional do video/filming. They don't zoom in and out like the amateurs do. There are a lot of cuts from different angles and distances, without zooming in and out.
This is a long way to tell you that you can save your $300, or use it to buy a tripod or gorilla pod if you don't have one. Upgrading a camera may improve your photos and videos a little, but improving your skill and style will go a much longer way. Try not to zoom in as much as you can. Use your legs to get closer to get better shots.