I don't know if individual chapters do it on their own, but the only Living Circle I've been to was at the 2001 Convention in Atlanta. There we sang songs (mainly the ones every Alpha Gam should know), at the conclusion of Convention. It was neat because after the Living Circle, some regional parts got together on their own and sang their own special AGD songs. I even have video of it! That's where the Hawaii delegation did our special hula.

The Living Purpose is another NM ceremony where our Purpose is broken down into parts. The narrator goes into detail on how we can incorporate those parts into our everyday lives. That should be in your VPMD/NMC's part in the binder. PM for details if you wish!