How can I get rid of this constant metallic taste in my mouth?

The cure depends on the cause!  Some medications cause a metallic taste in people’s mouths, as do some foods and sinus drainage due to allergies or illness.  Other times, the cause is unclear.  If you think your medication is the culprit, talk with your physician about changing to another similar medication, if possible.  Otherwise, see if you can determine what the offending food or drink is and avoid it, seek the advice of an allergist for severe allergies, or wait out your cold.