Follow Project Tempo

It'll sound better after you render the AS stretching permanent by bouncing to clips using an offline rendering algorithm (different algorithms for different types of material), but it will never be transparent, except possibly on some drum tracks and other percussive material with short transients. Try it on a copy of the project so you can easily revert. The tempo change might not be worth it it's not absolutely mandatory to make it fit something else.

I'm a relative newbie myself at this, and am trying to learn to do the same things.  I'm sure someone more experienced will correct if this is wrong, but try this:

If you still have the project open, undo those changes.  Now select each clip, and in the Inspector Clip tab, first make sure audiosnap is *disabled*.  Now open the Groove Clip tab and enable "Stretch to Tempo."  Leave "Follow Pitch" disabled.  See if that improves things. 

Also, I thought I'd seen a way to adjust the quality of the algorithm used to audition audio, but can't find it at moment.

An important aspect of this that I've realized only recently is that an audio clip can't be both a groove clip and an audiosnap-enabled clip simultaneously.  They are mutually exclusive.  In fact, I think groove clip, audiosnap, and V-Vocal are *all* mutually exclusive.  An audio clip can only be enabled for one of them at any given time.

Hope it helps, and I hope someone more experienced will expand/correct my suggestion.