Could be clarified a bit, but my take on the swap is that if 10 people are in the swap you send 20 stickers and end up with 2 of your own sent back to you.
If you were sending them out yourself you would only send 18 because you wouldn't bother sending 2 to yourself. (Therefore send 20 instead not 18 and everyone's envelopes are the same).
Therefore 1 envelope with a return your address and 20 stickers which can then all be collated into 10 equal piles to be put into the 10 envelopes.
You could also do the case swap thing and put in an extra sticker for the organizer.