[Addition to post #17] You can do it in hardware. Either use a controlled invert-non-invert amplifier with the inversion selected by the output signal, or invert the return signal so that you have both inverted and non-inverted signals and switch between them using an analogue switch (CD4016).

How would this clean up the signal for noise?

 

How would this clean up the signal for noise?

When you average the output, any noise that is present all the time appears at the output half the time inverted and half the time non-inverted, thus cancels out, whereas the return signal is inverted when the outgoing signal is inverted, so the two add up.

 

@Ian0 That makes sense. I presume this would be handled in the firmware. If the noise is purely random, averaging over a longer period should filter it out anyway. I can also adapt the firmware to observe a stable value over, say, 30 seconds before acknowledging it as a leak, rather than triggering an instant acknowledgement.

The wires connected to the ADC will have an RC circuit. However, the H ports will be either VCC or GND. What filtering can I add there to ensure that noise is not introduced via those wire connections?

 

This is a new design, and the aim is to keep all possible options on the PCB to minimise noise. I can therefore add a 10 kΩ resistor and a 1 µF capacitor.
At this stage, the H port is simply outputting 3.3 V and GND, and the polarity would then be reversed to produce GND and 3.3 V.
Please could you explain your suggestion regarding GND, as I am not fully understanding what you mean.

The effect of what the connection shown as "gnd" will have depends on what else it is actually tied to. If it is only a local common within your circuit there would not be much effect, while if it is also a connection to the realm outside of your circuit, that "outside world" will often include unwanted voltage variations. (NOISE)
Within this site I think I have seen a much broader discussion of "gnd".