Where could this green residue that is clogging fixture screens be coming from?
Green clogs in fine filters is usually due to algae being sucked up from the bottom of a holding tank. This is not uncommon in a water heater. This is a good reason to drain holding tanks annually to get rid of sediments. Also a good reason to never use water from a water heater to cook with. A water heater that is set below 130 degrees F is a breading ground for bacteria if not cycled often or allowed to set unused for several days at a time. (like when away on vacation) It could be copper oxide, but that would be very uncommon unless you have copper holding tanks.