My thought is that every seam stretched when you sewed, as all of the cuts were probably on the bias. What I would do is *heavily* starch the fabric before cutting the blades. By this I mean a 1:1 solution of Sta-Flo liquid laundry starch and water; spray starch is not nearly as heavy. My method for starching fabric is to "paint" the starch on with a large wall painting brush, wait a few minutes to allow the fibers to absorb as much starch as possible, toss in dryer, then iron with steam.
Also, when cutting the blades, I would place one cut side on the straight-of-grain of the fabric (the lengthwise grain is even more stable than the crosswise grain). I would also try to keep this always the same side of the blade -- e.g., the righthand side of the blade when rightside of fabric is up. This would ensure that you would always be sewing a straight-grain edge to a bias edge. (The edges most likely to stretch and distort while sewing are two bias edges together; having even one of the edges on the straight-of-grain will help.)
All of the above will help stabilize the fabric so it is less likely to stretch and distort as you sew. It is still a good idea to handle the pieces carefully.
I probably wouldn't do this, but you can also carefully trace the template onto paper so you have a paper template against which to place your plate as it is partially completed. It would help to catch distortions earlier, rather than waiting until all 20 pieces are sewn in place.
One last thing you might look at is your sewing machine. Some machines are better than others at sewing without stretching fabric. You might find that lightening the presser foot pressure helps (not all machines have this adjustment, though). Or, you might find that you need a walking foot on this particular machine. Usually heavy starching and careful sewing is enough, especially if you can cut pieces with one edge on the straight-of-grain.