THS4521-HT: Op-Amp giving negative differential voltage
The DC input offset voltage on the THS4521 is typically +/-0.2 mV and can be as high as +/-2 mV at ambient temperatures. Across the whole range it can be as high as +/-4 mV.

Given that you appear to have a gain of 20 this could result in an output offset of anything in the range +/-80 mV.

Maybe this is the problem? It sounds to me, on the face of it, that this device isn't suitable as a pressure transducer amplifier. I mean, it has a bandwidth of 145 MHz and, with a big bandwidth things are sacrificed in other areas (such as DC accuracy). 

Try using an instrumentation amplifier - there are plenty to choose from and I doubt very much (even under really strange measurement situations) if you need a bandwidth more than 50 kHz. There are plenty of IAs that are sub 100 uV input offset across a full temperature range. I understand you need a differential output of course and this can be accommodated by an inverting op-amp on the output of the IA.