Hi, Myke,
As you know that the Strain Gauges output is a differential signal, generally, it's output signal is very small, if you want to amplify it, you have to use instrumental AMP(triple AMP), which is expensive.
I suggest you use Kinetis KM family, the chip has two ADC module, one is SAR ADC module, another is AFE module, both of them support differential analog input and some channels of both ADC modules have PGA which can amplify the differential signal with programmable gain.
The AFE module supports PGA function and can test differential signal. It's rating resolution is 24 bits, but the actual resolution ENOB is defined in data sheet. the AFE module uses signal-delta technology, which modulate the tested signal to a high frequency signal, from theory, the technology can improve the time/temperature drift, the feature is important when you test a DC type signal.
This is the Kinetis KM family website:
ARM Cortex-M0+ Cores|Kinetis M MCUs|NXP
Hope it can help you.
BR
Xiangjun Rong