Yes it is true if you hand applique a quilt will become distorted, but if you machine applique it shouldn't be a problem. When hand appliqueing sometimes extra material is sewn under an applique just because you don't have your fabric flat at all times.
My suggestion is finding a machine applique, and yes, do it after you bind if you like.
Even though you are not free motion quilting the quilt you still can add excitement by just machine quilting a grid pattern, or straight lines, you could even do this with a colored thread and it would be quicker than applique if you are anxious to get the quilt done.