So I think I somehow got it to work. I uninstalled alsa-lib using the slackpkg tool, then I installed alsa-lib and alsa-driver from the alsa website. Without running alsaconf or anything, I rebooted my machine and everything works fine. You even get a nifty little beep!
The sound card alsamixer tells me it is using is: HDA ATI SB with Chip: Conexant CX20582 (Pebble). I also have a modified .asoundrc whose contents can be found in my other thread on setting up x120e with slackware 13.37.
Edit: So the above worked at first. Then after a reboot, the sound went all screwy. I then blacklisted a bunch of drivers and got it to work. I'm sure there is a better way to do this, but here is what I added to /etc/modprobe.b/blacklist.conf