I might have a similar issue. I am using AD8237 as a strain gauge amplifier (gain 500). The part seems to have hysteresis at small input voltages. If I pull on the strain gauge, I can see the input voltage change linearly increasing from about -.3mV,-.2mV, -.1mV,0.1mV etc. however the output stays at 1.0mV and stays there as I pull harder and harder until it jumps to like 20 - 30 mV then seems to respond to varying force from there. I am using the Ref pin grounded through a resistor equal to the parallel combination of the two gain setting resistors.
Could this be an offset problem as you detailed for Haox?