Current measurement opamp, is this circuit flawed? Output all over the place
Your input signal is 10A, which through your R19 sensing resistor develops a voltage of 10mV. Your opamp is specified with a maximum offset voltage of 7mV, which is 70% of full scale!

Note, calacuating the offset like this, it doesn't matter what your gain is (it's about x250), or your ADC resolution or scale, it's presupposes you have the gain and ADC correct to get a full scale measurement, and gives the offset in terms of full scale.

You have several options:

1) Calibrate the offset. The specified offset drift is fairly small, 1uV/C, or in full scale terms 0.01% FSD per degree C, which also fits with your description of the error being stable on any particular board.

2) Use an amplifier with a bit better offset, down to 1mV should be easily acheivable for not too much money with a bit of research. Offset down to 10% FSD.

3) Use a 'zero offset' amplifier, chopper or AutoZero based, which will be pricey, but will work. There are several options now from the big beasts TI and AD, which will get you 0.01% FSD offset.

4) Can you increase the size of R19? 10mV is quite a small burden voltage for an ammeter, though it could be all you want to budget for an in-circuit meter.