Why would I be getting dirty water from my bathroom sink faucet only when the cold water is being used? We are on a well if that matters?

Usually when you get dirt in water the pump for the well is too deep. Raising the pump about one(1) foot may solve this problem.

It would also be a good idea to install a full house filter where water comes into house.

Chuck

I also think it would be a great time to flush the sediment from the bottom of the hot water heater as well (something that should be done annually)... you know its not coming from the hot water taps because its settling in the heater after it comes into the house.

Assuming you are getting dirty water from outside faucets as well?

Is your pump in the well or above ground?