Are you quilting on a frame? My sister makes "wonky" quilts and I just started frame quilting a little over a year ago. What I found really helps is quilting closely (in my case, with Glide thread which adds a little sparkle and never shreds or breaks on me).
I did one quilt with Warm and Natural, one with Mountain Mist Blue Ribbon cotton batting (very thin and not needlepunched), and one with Hobbs PolyDown. I find the higher loft of the PolyDown more difficult to quilt on a frame without getting puckers; my favorite of the three so far has been the MM Blue Ribbon batting.
With all of these quilts I did closely spaced loops -- not more than an inch between quilting lines, and usually less. I did have to use soup cans on either side of the machine at times to distribute excess wonkiness.
I was very surprised at how well the quilts turned out, considering how many problems the tops had. I do think that the close quilting was what helped the most. I also think that using a high loft batting will probably create more problems for you rather than less. The highest I would go is the Hobbs PolyDown. I bet that wool batting would work really well; I will probably try that out next.
Incidentally, my sister's quilts are machine pieced.