I have a bunch of cores cut out for a sunset scene. I'm hand sewing the individual rows. When the rows are done, I might go to the machine to attach them. At this point, I feel like I have better control sewing by hand. Here's what I do to sew them.
Finding the center of each core is the important step #1. Fold the fabric and pinch it. Then pop a pin in to hold them in place. Then you can match the right side seams: the edge of the front piece comes up to the edge of the back piece.
Get the first stitch in the right side and give the fabric a tug. Not all fabric is equally stretchy. The weave may be tight and you will have some waviness but you can still ease it together and stitch it flat. The top edges will begin to line up when you gently pull on it. As they do, grab more stitches with your needle. You may only get to grab one stitch at a time or, if your edges stay lined up, you can get several.
When you reach the center, make sure your stitches are not pulling but are flat, do a little back stitch, and take out the pin. Let the right half curl around as you line up the left side edges. Pin that left side down. Repeat the sewing process for the second half and tie it off.
When you open the seam, you will see something like this. See which way the seams lie down best and press it. Then pat yourself on the back for sewing a lovely curved apple core seam!
Do you make projects that have curves? Craftsy has great lessons about it. Start small and simple, be easy on yourself, and give it a try!