This has probably been asked, but I looked and could not find it. In fact when I search it always matches on Pinnacle Studio, and the answer is to lock the audio track before doing a cut. Can't see how to lock audio independently in Avid Studio.

The best I've been able to figure is to split a clip, drag a portion to another track (which always seems to cause a slight time slide, which introduces a glitch in the audio!) and twiddle the audio out level on the two tracks. AS should be a step up from Pinnacle, so there's got to be an easier way to do it (albeit different) that I just don't see??? 

L-Cuts and J-Cuts are pretty basic stuff, one of the first thing learned in community TV studio classes.