Well, no, you should not be getting an Ubuntu desktop. If you have sound in Rhythmbox with an Ubuntu desktop, that might be OK, but if you are not, then that is not good at all. Do you have any sound at all from anything?
I was hoping that you had installed all of the necessary ALSA packages via Synaptic on your regular Mint install. I am unsure what is going on right now. As a last resort, you can do a clean install and start over. That takes well under an hour. I have done that a few times myself.
Perhaps someone who has more knowledge about sound - especially the Sigma Tel issue - will chime in and help. I have not had any sound issues before, but I can sympathize with your situation. I shall do some more reading and post back if I can find something useful. It may be a rough road with that sound card.
Have you tried a Live CD of another distro such as Mandriva? Mandriva One is pretty good. I have always had good luck with openSUSE KDE (4.3 GB DVD). If you can find a distro that Sigma Tel likes, that may be easier. Sometimes that happens. Good luck.