Well this would certainly depend on how one considered the immunity.  The body has a lot of areas to resist harm aside from just the liver.  For instance some medication is processed in the kidney and the lymph node system handles much of the fighting of disease and illness.  Alcohol works as a depressant on the nervous system and is carried through the bloodstream to those receptors.  Continual ingestion of alcohol would probably allow the pair to get drunk as their healing factors do take time to work.  Their euphoria may be short lived though as their body moves to reestablish those systems, but continual ingestion of alcohol could probably keep them "ahead of the curve" so to speak.