Hi Dave. Not sure if this could be anything, but mostly I2C EEPROMS are used to store CAL data and constants. Some would remain same over the years, some might change depending on a lot of factors.
Your 5V (digital) rail is OK, so its unlikely the I2C EEPROM has failed, however have you considered the possibility that when the original PSU went kaput, it might have caused a high spike or kickback, and probably that was enough to either : a) destroy the I2C EEPROM itself, OR b) corrupt / destroy some of the CAL data on it ?
From a software point of view, the software would start by getting the CAL data from the I2C EEPROM and then start reading the ADC, presumably by connecting one or all channels to known reference voltages to perform calibration. Most of the times, you need to have the previous CAL data (and that too within close limits) to get that CAL function to return a proper value. Otherwise, if the return values are not within a specific limit, it MIGHT just assume that the Analog or ADC section has failed !
Just a hunch.