Well, I found this which describes my symptoms fairly well.

http://www.ipm.ucdavis.edu/PMG/r261100411.html

Doesn't sound like anything practical can be done. I take for granted all the good things that have to be in place (and bad things that need to be absent) in order to get those delicious fresh figs. It's truly a blessing to have a nice crop. Maybe next year!

May be you do not have a Celeste,but other ,mongrel hybrid,fig,that has an open eye.
In Ca,Ficus Carica ,cross pollinate with Caprifig,and ,can be have fertile seeds.
A hybrid of Celeste(random),looks like Celeste but ,has a larger eye,or an interior cavity,and will sour.

Same thing here. First time and I have lived in the deep south all of my life. It always rains and it is always hot and humid. Still it doesn't stop the birds. They are having a great time raiding my tree. By the way, I have brown turkey.

The tree I have in CA is a transplant from a Celeste in La. It made good figs for about 8 years until the problems I describe above.
Next door to me is a 200 acre plot that operated as a nursery in the years I had good figs. Two years ago that field was planted in strawberries which start to yield in Jan, about 8 months before my figs ripen. I'm thinking it's possible that all that fruit next door is hosting pests that make their way to my tree and cause this spoilage.