I’d add a fourth guess to Larry’s three.
From the way you describe it, I think it’s aluminum anode corrosion byproduct. Stuff in your aerators could easily be disintegrating dip tube, but the fact that it makes your laundry dingy makes me think you have a water heater with an aluminum anode. Tell me the make and maybe I can confirm it.
Solving the problem may just be a matter of replacing the aluminum anode with a magnesium one.