Wrong voltage in op amp circuit to adc
Wild guess here but 5K is not a standard E96 value - are you sure it's not 4.42K, which is a standard value?

Anyway, assuming that's not it- to debug it, read the voltage across the shunt with a voltmeter (preferably with 0.1mV resolution or better and >10M input resistance). Most inexpensive handheld meters are fine. If it's not 78.0mV +/- shunt tolerance (measured between the sense connections, not the force connections), then you have at least one problem. As in a comment make sure you've used a 4-wire type shunt and are using it correctly (though problems here would tend to lead to a higher output). 

When that's okay, read the voltage across the op-amp inputs. It should read 0.0mV. 

If that's okay, disconnect the ADC and read the voltage across the 5.00K resistor. It should read 1.30V +/- all the tolerances (taking Vos of the op-amp into account), plus a few tenths of a percent for BJT hFE. 

If that's okay, then it could be loading the by the ADC. Some ADCs have input impedance in the range that would explain your symptoms (~38K).