Instrumentation (or diff) Amp saturating when measuring low voltage
A TL082 is somewhat inappropriate for low-level instrumentation- the offset can be as bad as 20mV and the drift is not guaranteed but typically is 10uV/°C. A gain of only 1000 means that a 10mV offset will saturate an amplifier with +/-10V supplies, even with 0V in.  

If it was 1975 and you really had to use something that bad, you could put an AC voltage on the bridge and measure the AC voltage coming out of some AC-coupled amplifiers. 

However, it's not 1975 and we have access to chopper-stabilized "zero drift" amplifiers that have offsets in the low uV and drifts measured in tens of nV/°C. 

As an alternative, and since you're using a microcontroller, you could use the micro to alternate the voltage across the bridge and do synchronous detection, which is more like the first method, but involves a bit of firmware. Input noise voltage is 25nV/rt-Hz in the white noise region but you may not be able to achieve that, depending on the 1/f corner frequency. 

Depending on how you want to proceed, you should pick one of the above methods. It's not entirely clear from your question whether you're trying to measure to a precision of 20uV or whether your entire signal is 20uV and you wish to measure it to a much higher precision. I assume you're actually looking for a resolution of 20uV, which is not that difficult to achieve in practice.