Have you contacted your local water company to do an analysis on your water? usually, they'll either send you a kit in the mail or will come take a sample for free just to make sure there aren't any issues.
You can't go wrong with flushing the hot water tank.
We have horribly hard water here in Indiana and have to use a water softener, too. We prefer Morton's yellow bag salt over others - get them at Sam's Club much cheaper than elsewhere.
We learned that our water softener can't keep up with the volume of water in our house, but it was going to be $4000 to get one that could, so we deal with the lime residue - I've never had problems with nails, clothes, etc.
It would be concerning to me - partly because I'm a cancer survivor, and partly because I've never experienced what you're mentioning with your softened water.
Good luck - hoping you get it resolved soon.