Why am I getting water scaling even with a softener installed?
As a guess, the pipes got layered internally in scale, that will come out over some time (dissolving into the now softer water). Some large chunks may require filter screens, shower heads and valves to be soaked in your vinegar or CLR (containing lactic and gluconic acids).

I know boilers get descaling treatment, but I've not heard of the same with standard pipes. If they were quickly descaled, there might be a large amount of debris, which would necessitate disassembly (washer inlet filters, refrigerator water connections, shower valves, toilet valves). Perhaps all the hose hookup devices could be disconnected for the treatment