Actually, I'm a Contractor, not a chemist, but MY IMPRESSION is that your water is too hard.
Try these things: 1.) People tend to use too much detergent in their dishwashers. This does three things: it wastes the detergent, it etches the glass in your glassware, and it makes proctor and gamble's bottom line a little better.
2.) Turn your water softener back on, then have your water tested. (for all I know, you may be able to do some kind of test at home, but I've never had to deal with hard water where I live, so I don't know for sure)
Your comment that you still have residue on the dishes when you hand wash them confuses me. Are you hand washing dishes that have already been through the dishwasher? Are these "clean" dishes that become powdery when you hand wash them in the sink alone?
You haven't isolated the source of the powder. It could be the water (my guess) or it could be too much detergent in the dishwasher, (common, but again, you are confusing me here.)
My best guess is your water softener isn't adjusted correctly and is either putting too much salt in the water, or is not softening it enough. My guess is not enough, because you didn't seem to notice a difference when you turned it off.
Call a local repair company who services your brand of water softener and ask them your questions. If you determine it ISN'T the water, then call your dishwasher company or the soap manufacturer. GE's Customer Service Number is 1-800-626-2000.