Is there any way to finish this game without killing the Shriekers? I have hit them multiple times with two different purging wands (neither one of them inert), to no effect. I know that there is something about Vampirism Source but I have not run across it.

If the shriekers at the end of Act 1 are bugged out, then you'd have to stock up on invisibility sources, then chug one for the next character in line to sneak past - and take the left path (central heading straight for Alexandar, right heading for the sneaky treasure crate) and teleport everyone up from this spot:
https://dl.dropbox.com/s/rgcex5wk0c7k3v1/sneeky.png

There's a way down from there, you'll just be starting your encounter from a highground choke instead of in the open like this.

https://dl.dropbox.com/s/2lxbuk02nn3cpsz/sneeky2.png

Edit: confirmed that you can indeed drink an invisibility potion (chameleon cloak would also work, if you have it) and just walk past the shriekers, then get up and start the fight from said highground position

You can also free the dragon Slane and he will kill the shriekers for you. If they still don't die because of bugs, then just try the sneaky method =/

IIRC, the only way to kill the shriekers is to purge the source pools which feed them--aiming the purging wand at the shrieker on the cross has no effect--you must aim the purge wand at the bottom of the pole where it can purge the source and kill the shrieker. This is from memory, so pardons if it is incorrect--but I remember at first having difficulty killing the shriekers so I went back and read what material and dialogue I had collected on the subject and figured out from that how to use them properly to kill the shirekers. Purging wand works fine but you must know where to aim it...;)

To kill a (non-bugged) shrieker you simply use either Source Vampirism that you acquire in the middle of Act 2 or Purge (which is the same ability) while highlighting the hanging shrieker unit with the ability, as you'd do for any other ability or unit.