If your cold and cough have lasted a week with a dry cough and frequent spitting, it is likely a lingering viral infection or throat irritation. You can try warm fluids, honey with ginger, steam inhalation, and saltwater gargles to soothe the throat. For relief, a cough syrup like Dextromethorphan (for dry cough) can help. Stay well hydrated and avoid cold or dusty environments.