It sounds like your hard water is corrosive. Softening is preventing hard water scale from forming on the inside of the pipe and the softened water then dissolves the scale and the water still be corrosive will start to corrode the pipe. This will add copper to the water and when the water evaporates the blue/green copper will stain fixtures.
This is not caused just because you softened the water. The water was corrosive in the first place and was also scale forming. The hard water scale protected the pipes. I suggest that you look into a Neutralizer. This unit looks like a water softener but does not have a salt tank. It should be installed ahead of the softener. This will neutralize the corrosivity of the water. It will add some hardness also so your water softener will have to be regenerated more frequenty. You can contact a local water softener dealer who will likely analyze the water and recommend the neutralizer and will also adjust the regeneration frequency of your softener.