Sorry for the late reply, really busy lately, actually started already in mid January and now ... hell just broke loose with this Chinese bat-shit virus
I was suggesting to try the Live version because it has an actual kernel (drivers) + pulseaudio + alsa.
Don't know why the Live works and -current doesn't, both should use the same latest code (the above packages). - @Alien Bob (Eric) could help here.
I would investigate (compare) if there are different/additional kernel boot parameters, the pulseaudio conf (/etc/pulse/) files and maybe how udev is launched in /etc/rc.d/rc.udev together with the rules in /lib/udev/rules.d/
P.S. Wait, I believe I went too far with the above recommendations and haven't focused on the actual kernel you were booting with the Live Slackware.
I don't have too much experience with the Live image and also too lazy to parse the build script(s). This help page isn't really helpful in identifying which kernel is used by the Live image by default:
https://docs.slackware.com/slackware:liveslak
But I believe Live uses the generic kernel and -current is using the huge one by default. There could be some differences in Sound HW support between the two. Never used the generic kernel on my own, can't tell the difference.
You should boot Live, check what kernel you're running generic/huge and use it also in your -current installation. Also, run a dmesg and paste the section where your Sound HW is identified in the Live Slackware. That will help in identifying if there are some kernel drivers/modules missing in one of the kernels and can be supported (enabled) in the next -current kernel release.