Protect Op Amp input from piezo sensor overvoltage
What's confusing is that the + input of the LM358 (above) should be grounded to mid-supply.

A charge amplifier works by keeping the inputs to the op amp at the same mid-rail ground voltage.The output from the sensor is a current signal which charges the feedback capacitor. There shouldn't be any voltage signal directly from the piezo sensor, it should just be current. The integrator converts its input current to output voltage.

I think what's happening is that when the output from the op amp integrator hits the rails, its inputs move apart and exceed the power rail voltages which the op amp won't cope with.

The maximum input for a LM358 is 32V but the minimum limit is Vss-0.3V.

Your two zeners (5.8V limit) will still allow the input to exceed the negative rail voltage. (9V battery).

I think I would try putting a couple of ordinary 1N4148 signal diodes back to back between the op amp's inputs to limit the voltage swing of the op amp's negative input to 0.7V either side of mid-rail ground. The added diodes shouldn't affect anything until after the integrator's output hits the rails.

You could then confirm, by scoping the integrator's output that it is still hitting the rails proving that the piezo's output is indeed a current signal.