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I have been afraid to raise the ISO too high because of introducing noise. Also, often I cannot use a tripod because I am following active children or pets around the house. As a result, I have tried to keep the shutter speed at at least 1/50. I use TTL with the SB600. Again, I would appreciate any further advice that you might have. I hope I have explained things better now. Thanks so much.
You're imposing too many limits for the camera to be able to achieve a 'perfect' exposure for the dim ambient light.
If you insist on having a low iso AND a fast shutter speed, the only variable you've given yourself to play with is the aperture. Once you're at f/2.8, you can't go any wider, unless you use a prime lens with a wider max aperture (eg. 50mm f/1.8). Hence your camera is simply not capable of satisfying all the conditions that you have imposed.
If there is limited light (ie indoors), you need to relax those constraints. You have a few options
1. Set the iso higher. Modern SLRs are capable of very good results at high iso, SLRs are much better than point-and-shoot cameras. Try setting the iso higher and see if you're happy with the results. I'd rather have a (slightly noisy) clear picture than one which is blurry because my iso was too low
2. Set a longer shutter speed. The flash will freeze your subject, and the longer shutter speed will allow the background to register in the picture. This works best if your subject is in a dark part of the room with a brighter background, as it relies on the flash providing most of the light on your subject; any ambient light on your subject will show up as motion blur around your subject. Set your flash to '2nd curtain' sync. Motion blur can give some nifty photo effects though, many wedding photographers use this technique on the dance floor
Or you can just set your camera to auto with the flash pointed at the ceiling and see what happens. The Nikon flash system is pretty clever, make use of it.
Hope this helps