Kathy,
There are several ways of doing applique. The blanket stitch method your grandmother used is still used today. There is no seam allowance added to the template. You may want to spray starch your fabric before cutting out it give a bit of stability. The blanket stitches are done to match the color of the piece being appliqued or you can do all in a contrast thread (2-ply embroidery). Black is often the color choice to do this. -- Some folks will fuse to the background fabric. It gets a bit stiff which is not appealing to me.
Some sewing machines have a blanket stitch that is often used to achieve the same result. Obviously, it is much faster. If you choose this method and come to curves, you will need to stop every stitch or two, reposition and continue stitching. Also, practice doing a few points before attempting it on your good project. They are tricky.
Blessings. Suzanne