What can I do when black sediments is coming out of my bath faucet . I have well water the sediments are coming out from both cold and hot water?

Do you have a water softener? Sometimes the diaphram splits and exchange resin gets through. If you have a carbon filter in the whole house filter it may have ruptured. Check the substance with a magnet. If magnetic it could be black iron oxide.

Pump or city water" One handle or two knob bath faucet? Are these particles hard or soft?
Let me know, Tom