Welcome to the group! Unfortunately Mr. Ripper is a best friend to most of us!
If the edge of the fabric is getting caught in the feed dogs, you may be making your seams too "scant 1/4"...something I did as a beginning hand piecer/quilter.
Try making them just an oooch (is that how you spell that???) wider and this may take care of your problem.
You do need to be sure your pieces are cut straight to the grain of the fabric for best results...straight edge of cut lining up with a thread of the fabric.
You can always use Fray Check, but that could get kind of pricey pretty quick if you used it on all your pieces.