You sound like me when I first started. Yes you can use a glue stick to put them together to make a larger piece. Just don't get too many layers in one spot. I have a plastic box that is my collection of scraps. One thing I have done, is if I am using a roll of stabilizer I don't trim it off the roll until after the embroidery is done. Instead of stick gluing I use the spray adhesive Sulky KK2000. It is a bit pricey but it releases ofter 20 minutes or so. And more importantly it does not stain like another one I used to use did. So if I don't have a large piece that may or may not be caught in the hoop, I spray the stabilizer and have the machine do a "fix" which holds it in place the rest of the time.
When using the water soluble I just wet my finger and run it along the edge of the left over and it adheres to the other piece.
Pieces of stabilizer will multiply just like fabric scraps - the more I use them there seems to be more. (The closet where I store them is dark so who knows what happens when the door is closed.)