Thanks for the advice! I must admit I was thinking mylar and plastic was the same thing. I didn't realize that there was a separate plastic template material at all.
I will try the three layer freezer paper, as well. I really have no common sense in quilting at all! I was tracing two separate templates on freezer paper, cutting both out separately, and then ironing them together, so of course there were probably small differences that weakened the finished product! However, thanks to all the very kind advice on QB, I will keep at it! (Slow, but determined!)
I will also be ordering the Templar template plastic this week! I have high hopes for this, as I really like the stiffness and endless re-usability.
Here's to conquering applique one tiny stitch at a time!