Separation anxiety is very treatable and very, very common in shelter dogs when they first get out of the SPCA. Dogs are pack animals and look to their leader for guidance and direction. Dogs in shelters are seen by many people, and dogs don't know who their leader is. Many are scared at an SPCA shelter. When they get into a foster home, dogs need time to adjust to these new surroundings.

I fostered many dogs for the city pound and every dog had some separation anxiety that always went away within a few weeks to a few months. My dog, Holly experience probably the worst seperation anxiety and that was fairly easy to treat. I would give her a bone or something really wonderful when I went out (I did not put her in a crate..this is scary too!) And I put anything I valued up high that she could not reach (one could always baby gate sections of the house to keep doggy away from any valued furniture). As long as Holly had a chew toy, or bone and didn't see me walk out the door with my worried face watching her, she was fine.
It is a terrible, terrible excuse to kill a dog due to seperation anxiety. May poor Lucy rest in peace now.