I've been Winterizing all types of RV"s for over 35 years.I do several hundred every year & it's unbelievable the junk that comes out of the systems.
I would backflush the system with good city water pressure. You will need a few adapters, one that goes from areator thread to hose thread, a female hose to hose swivel fitting & a 1/2" male pipe to female hose.
This will allow you to connect the male end of a hose to your kitchen & or bath faucets. the male pipe to female hose will allow you to hookup to your shower hose and or toilet line. I would remove the plug from your water heater (making sure it is off, both gas & electric)You can then go to each fixture and blackflush both hot & cold to the heater drain of the water heater.Do the farthest fixtures from the water heater first.
When you reactivate, do the tub first since it doesn't have an aerator anything still loose should come out there. (I always recommend this when you first hookup anywhere)
If you think it's coming from your fresh water tank, install a good filter on the pump, Shurflow has several.