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It sounds like you're describing mineral deposits in the water that precipitate out in the water heater due to the heating of the water. Then they get swept up into the tub when you run the water. Are you on a well? or does the town water supply come from a well?
Does your town get it's water supply from wells or a reservoir?
Have you called the water department about this? The only likely source is the town's water. There's nothing in the plumbing system that would/could produce what you describe.
Dip tubes are plastic. What you describe doesn't sound like plastic. Not to mention the problems you would have getting adequate hot water if the dip tube was failing. You probably don't get this through the other faucet because the flow into the tub is far higher volume that anything else in the house. So sediment is more likely to be noticeable there.