Your ADC isn't working because you're exceeding the ABSOLUTE MAXIMUM RATINGS listed in your question:

Analog inputs to AGND: −0.3 to AVDD + 0.3 (V)

In other words, this ADC cannot measure negative voltages. Most simple ADCs can't do that.

Your ADC isn't working because you're exceeding the ABSOLUTE MAXIMUM RATINGS listed in your question:

Analog inputs to AGND: −0.3 to AVDD + 0.3 (V)

In other words, this ADC cannot measure negative voltages. Most simple ADCs can't do that.

You could overcome the problem by adding a constant offset voltage to your signal (google "Non-Inverting Summing Amplifier"), then subtract this known offset from the ADC measurement in software. Or, if electronics is not your strong side, you could simply put the signal through a resistor divider attached to AVDD: a 2x divider will let you measure voltage from +AVDD to -AVDD:

^{simulate this circuit – Schematic created using CircuitLab}