Audacity uses the OSS drivers. AFAIK, programs which use these drivers want the soundcard all to themselves, so if you have any other programs which use the soundcard you have to close them.
The alsa oss thing sometimes works to make OSS programs play nice with others, although I think this depends a lot on your soundcard. Personally I've never had much luck with it.
The most likely program using the soundcard is some sort of sound server. I'm not sure about slackware, but I know lots of distros enable these damn things by default, causing no end of headaches like the one you describe.
In gnome you can disable the sound server in the System -> Preferences -> Sound -> Enable software sound mixing (ESD) [checkbox]
In KDE you can disable the sound server (if one is running) in the System Settings panel -> Sound & Multimedia -> Sound System -> Enable the sound system [checkbox].
Disclaimer, both these config panel instructions are for Ubuntu Dapper, your distro may vary!
Other notable programs which use OSS:
Skype < 1.3 (The new 1.3 beta uses ALSA - yay!)
Enemy Territory and many other games
Flashplayer 7 (version 9 - when it finally arrives will use ALSA apparently)
It's amazing anything still uses it. OSS has been deprecated in the kernel for donkeys years, but somehow the message didn't get out to a lot of people.