Measuring output of op-amp current shunt circuit using ADC
The biasing approach is what you need to do - offset the signal such that a zero input is somewhere near to the mid-range of the A-D scale. You then subtract the reading you get with a zero input from the A-D during measurement, and have a signed value resulting.

You can use one of your op-amps to introduce the offset.



You have a shorting relay across the input you can use for the zero measurement, but I'd suggest you use the other contacts on that relay to clamp the signal down away from the shunt, or you'll be measuring the contact resistance if there's any current present.