I haven't heard the term "pillowing", but I'm guessing you're referring to the stretched out fabric. Usually that's caused by one of 3 things:
Fabric isn't cut on on straight of grain.
Fabric is stretched during sewing
Seams are not consistent size
If it's a pre-cut, one would hope it's cut on the straight of grain, but it isn't always. If you catch it before you start sewing, you might see about exchanging it.
To prevent stretching, I heavily starch my fabrics before I start sewing. Sizing also works well.
To get a consistent seam, you can use a seam guide, a 1/4" (patchwork) foot, or a piece of tape on your machine bed to mark 1/4".
At this point, I'd suggest trying to press (NOT iron) the top with a lot of starch or sizing. It won't be perfect, but it might be good enough. To square it up, after pressing, fold it in half left to right (making sure the fold lines up with the center of the heart) and cut the 2 edges; then fold in half the other direction & do the same.
PS- I think the darker rows give a nice ground for the quilt. Maybe live with them overnight & see if they still bug you. Sometimes we can be our own harshest critics.