Hi Andrew,
I'm sorry to hear that you are still having issues. The fact that the replacement device initially worked and failed over time is probably due to issues with the prototyping system. Unfortunately the ADS1248 is quite delicate with respect to ground. Protoboards are notorius for oscillation and noise pickup. It is not hard to exceed one of the Absolute Maximum Ratings.
When prototyping a system prior to constructing a PCB board it is better to use a proto board where components can be soldered and wired connections are also soldered and wired with short connections. The traditional white plug in breadboard has lots of inductance and capacitance. It is possible to use, but most people (including myself) are in too much of a hurry to wire the board correctly. What I'm referring to here is trimming all wires to the correct lengths and making sure that they do not cross over one another. All components used should also be trimmed to length, especially capacitors. Using component leads for connection to power supplies, scope probes, volt meters, etc. is not a good idea.
If you have loops of wires and components with long leads, don't expect to get good performance from a 24-bit device. If nothing more, the noise will be too high to get any type of decent performance.
Best regards,
Bob B