Negative to Positive Voltage Converter
I'd use an LTC1043: -



Basically you can sample a voltage that is negative on pin 13 and 0V on pin 7 then, when you toggle the device the charged up capacitor gets rearranged so that the neg terminal of the cap is connected (via pin 14) to measurement 0V and the 0V terminal of the cap is connected (via pin 8) to the input of the ADC.

In the diagram above it might be a good idea to keep the LTC1013 to act as buffer to your ADC.

Another issue is that you'll need to provide +5V and -5V supplies to the chip AND your signals will need to be scaled down to 5V before the differential input.

Do you also see that you can use two LTC1043s to measure +V and -V voltages - operate them inverted - whilst one is presenting a voltage to thE OP-AMP the other can be sampling.