Apart from the increased noise floor the circuit still works (sensor signal visible in CHG_AMP_OUT)
That some signal is visible in CHG_AMP_OUT does not mean that this signal can be used for measurements.
Using your "Sensor Events" graph, I've tried and "reverse-engineered" your piezo sensor. To comply with your ca. 50 ms of output overload recovery time, the injected charge should be on the order of 3uC with the sensor capacitance of 15nF:
The voltage at the opamp's inverting input is ca. 50 V at maximum. You can argue that your sensor capacitance (which you do not specify in your question) is different and also the parameters of model's voltage source pulse (amplitude and timing) should be different, but all these values are constrained by the requirement to have an agreement between the simulated waveform and your "Sensor Events" graph. You can verify with simulations that, with any realization of a sensor model giving a simulated waveform similar to your "Sensor Events" graph, you still have the voltage at the opamp's inverting input on the order of tens of volts or even greater. As this value exceeds the supply voltage by more than 2 V, output phase reversal is unavoidable (AD8605 datasheet, OUTPUT PHASE REVERSAL):
Phase reversal is defined as a change in polarity at the output of the amplifier when a voltage that exceeds the maximum input common-mode voltage drives the input. Phase reversal can cause permanent damage to the amplifier; it may also cause system lockups in feedback loops. The AD8605 does not exhibit phase reversal even for inputs exceeding the supply voltage by more than 2V.
and your plot supports the output phase reversal hypothesis.
A signal conditioning circuit must do its work of conditioning the input signal, which means, among other things, scaling the signal to a predefined operating range. If the signals of your "sensor events" are typical for your application, your signal conditioning circuit does not "condition" signals at all.
If these events are not typical but very infrequent (outliers), at the very least you should protect your circuit. The AD8605 datasheet instructs you on input overvoltage protection:
The AD8605 has internal protective circuitry. However, if the voltage applied at either input exceeds the supplies by more than 2.5 V, external resistors should be placed in series with the inputs. The resistor values can be determined by the formula
$$
{{V_{IN}-V_S}\over{R_S+200Ω}}≤5mA
$$
The remarkable low input offset current of the AD8605 (<1 pA) allows the use of larger value resistors. With a 10 kΩ resistor at the input, the output voltage has less than 10 nV of error voltage. A 10 kΩ resistor has less than 13 nV/√Hz of thermal noise at room temperature.
With the input voltage Vin = 50 V, the series resistor RS = 10 kΩ satisfies the datasheet formula. I must confess that I don't know where the parameter 5mA come from: it cannot be found elsewhere in the datasheet except this formula.
Expectingly, introducing these protection resistors into the simulated circuit does not change the simulation waveforms. Also, the AD8605 SPICE model does not predict output phase reversal. So, we can only believe in the datasheet instructions and test the PCB with the recommended resistors in series with the inputs. Do not expect that these resistors make the circuit suitable for measuring outlier signals; these resistors only provide protection against the input overvoltage up to 50 V.
As for the "noise" arising as the result of overvoltage events, it might appear to be unexpected and even enigmatic, but you cannot expect well-defined behavior when using devices outside their specification parameters, and 100 nF capacitive load for TLV900x voltage follower is well outside 500 pF. The datasheet only promises that the resistive open-loop output impedance makes stabilization easier with much higher capacitive loads but does not guarantee that it does not require some effort. If, irrespective of your sensor problem, you are interested in stability compensation measures, read about out-of-the-loop and in-the-loop compensation techniques. For the time being, follow the advice given in the comments and answers to get rid of the parasitic oscillation problem in your circuit.
