Negative differential to instrumentation amplifier causing problems
Your output voltage is relative to the Vref pin.

If you fed the amplifier, ADC and strain gauge all from 0v and 5v, then taking Vref to mid rail would give you an output about mid rail, nicely in the 0 to 5v range of your ADC. 

That amplifier only gives you 5x gain in its default configuration, that's not a lot to follow a 16mV swing. Use the Rg terminal with a pair of reistors as shown in the data sheet to increase the gain.