Don’t know if this will help but.. (and I assume the tree was living a few weeks ago).
What was the diameter? The rule of thumb is to leave the walls 10% of the diameter. You may not be able to do this due to the small diameter of a cup.
You stated you shaped the outside. You must also shape the inside leaving the walls 10%. When I have tried cups/goblets it has been with dry wood. Some people do turn them green/wet but be prepared for warping.
Did the piece still contain the pith? If so it is very difficult to keep it from cracking/splitting.
It probably happened due to a lot of factor, the largest being the wood was not dry (at least <12% or so). When green it was probably in the range of 25-30%.
Many people (myself included) bag the item in a paper bag. I have never used the shavings in with it but a lot of people do. You need to coat the end grain with a sealer, such as wax, before bagging it. You are trying to control the moisture loss. The bag allows a slow moisture loss while keeping any air flow away.
I would think a cup would dry in a month (1/4” thickness).
I have picked up green wood and had cracks appear by the time I got home (thirty minutes or so).
They who can give up essential liberty to obtain a little temporary safety, deserve neither liberty nor safety. Benjamin Franklin