How to level shift + / - 1V waveform signal to 0 - 5V
The simplest way is to generate an artificial 2.5 volts pseudoground, then connect what are currently the ground points for R5 and R2/R3 to that. This way, the AC coupling that you are already doing will be to a nominal zero level of 2.5 volts rather than 0. If you do this, though, you will definitely need to replace the TL071, since you will be really violating common-mode input limits. Without knowing your packaging limitations, I'd suggest that you simply add another battery, and operate from +/- 9 volts.

Also, I suggest you look closely at pages 4 and 8 of the AD620 data sheet. You will notice that with increasing gain the (effective) input noise actually drops, so using as much gain as possible in the first stage is (as always) a good idea. I'm not sure what "noise saturation" means, but I'd recommend putting a pair of matched 1000 pF caps from the 620 inputs to ground to act as a low-pas filter for noise. As a matter of course, you should ALWAYS add input capacitors to any high-impedance amplifier which is connected to the nasty outside world. While this will, of course, decrease bandwidth, unless you absolutely, unequivocally must have that bandwidth you're best off without it. James Roberge, once a big name at MIT, had a saying: "People who ask for more bandwidth than they need deserve what they get."

Another reason to increase the gain of the AD620 is to decrease the effect of the second stage input offset errors. Assuming you run R1 at 50% (500k) your DC gain will be as much as 10 mV, depending on which TL071 model you use. 500 x 10 mV is 5 volts, and this is clearly more than you can afford. More gain in the first stage means less offset in the second, and the AC coupling will get rid of any DC offset from the first stage.