Alice,
If you were in Australia, I would tell you to get a 1lt can of Fungishield (wood stabilizer) and soak the wood with it, then put it to dry in a dark and well ventilated place. Any repairs/fillings could be then done without any problems...!
Knowing that you don't have this product, and unless you can find one that has identical characteristics, over there, one of the easiest ways to try to stabilize the wood, until it dries enough for the repairs/fillings, is to submerse it in soap water for a few days. The same dishwasher detergent you use is good and cold water in a container big enough to accommodate the bowl fully submersed.
Another method is to boil it by, submerse it in hot/boiling water for a couple of hours...!
Also, depending of the bowl's size, you could microwave it with "bursts" of maximum of 1 minute at the time with the wood cooling down in between bursts. Weighing the bowl before putting it in the microwave, is the best way to find out when is dry, as the wood will stop loosing weight, then it does...!
From this point, you can fill/repair the wood and completely finish the bowl, as the wood is now dry...!
Good luck...!
PS: Remember that with green/wet wood turning, and unless you want to take the wood to the final dimensions to allow the wood natural behavior (with most woods) of "setting"/drying out of shape, green/rough turning, as they call it, you need to leave about 1" of wood thickness all over the piece, this will give a little extra support to the wood while drying and also, a safe thickness (in most cases) to allow you to re-mount the piece and turn it to final shape, removing all the out of shape bits...!
Cheers
George
