Fishman,
Anemones are very... "tender"... for lack of a better word, animals. They require an added acclimation time, as well as pristine water conditions to survive happily. Usually, receiving one that is already exhibiting a "tearing" is a sign of it being very stressed, and quite possibly on its way to a not-so Happily-Ever-After land.
I would recommend you test your water. If needed to a 10-15% water change... and keep your fingers crossed. If you have clowns that are pestering it, this will also add to its stress level greatly as it is acclimating.
As for the splitting. They do often create smaller "siblings" when they split. So its possible this is what you are seeing in your overflow. Normally to help them split, they will find their way directly in front of a powerhead, return line... or as in your case an overflow... and I'm assuming they use that added flow to help in their splitting process.
If this "sibling" is only a split, you will want to tenderly return it to your main tank so it has a better chance of surviving.
I wish you the best of luck with your new addition. Keep us posted please.
Ed