Saturation when using differential amplifier with op amp
The NIOSH states "Under dry conditions, the resistance offered by the
  human body may be as high as 100,000 Ohms. Wet or broken skin may drop
  the body's resistance to 1,000 Ohms," adding that "high-voltage
  electrical energy quickly breaks down human skin, reducing the human
  body's resistance to 500 Ohms."


Let's say that body resistance is 100k and the op-amp chosen is a TL082. This op-amp has an input bias current of 50pA and through 100kohm will generate an offset voltage of 5uV. Multiply this by the gain of 1000 and no problems - the output voltage will be 5mV and not end-stopping. However, the TL082 has an input offset voltage of 5mV so this would offset the output by 5V when gain is 1000. But remember that two op-amps are present and their offset voltages could be additive.

What about the LM324? Bias current is 40nA and with 100kohm this generates an input offset of 4mV and not surprisingly produces a whopping 4V offset after amplification of 1000. It looks like the beginning of the problem if the LM324 is used but there's another issue - the LM324 has an input offset voltage of typically 2mV so, after a gain of 1000 this might look more like 2V added to the 4V due to input bias currents. Given also that there are two amps the offset voltage could be additive making a total of 8V offset.

An LM741 has a bias current of 80nA and this, given the same rationale as above produces an 8V offset on the output. Plus, the input offset voltage is maybe 2mV thus adding another 2x 2V. Grand total is 12V offset.

My favourite quad op-amp (OP4177) has a typical offset voltage of 25uV and a bias current of 0.5nA - this would, in the configuration above produce 50mV at the output due to bias currents and 2x 25mV due to offset voltages. 100mV total offset, ahhh that's better....

So, what op-amp are you using?