Multiple ADC readings in Arduino Uno with INA125 problems
I don't see any obvious reasons here, but, you talked about low amplitude and high gain amplification and this is where good decoupling, correct grounding comes into account.
It could be that your problem doesn't come from a schematic (or principle) level but from a layout problem.

If this is done on a breadborad and not a PCB, make sure that your decoupling caps are very close to every power pins of your amplifiers.

The observed coupling between the INA could come from the power lines or the ground line.

The added capacitor will decrease the impedance of the signal that enters the ADC, which is good. But the value chosen is HUGE and may present very high ESR. IF your ADC samples at 500 KHz for instance, your capacitors won't be very effective at that speed. This could explain the noise you see.

Another source of the noise at your ADC is the wiring. You have to ensure that the signal and the GND are very short between your board and the arduino.