silent screamer: many here have good ideas, Dr. Acula wrapped it up pretty well. To understand accurate shunt use and why the diff amp is so valuable,
consider expensive 5 and a half digit DVM's, like a Fluke or HP. For up to 10A
you can use the jacks on the front, going to your load, to measure DC current.
But for more accurate results, eliminating the voltage drop in the leads which is an error, you can use a Kelvin conn. All this means is that there are 2 additional leads, the sensing leads, that clip directly across the load or shunt. These sensing leads DO NOT have the load current going thru them, so there is no voltage drop error in the measurement. That is what we are trying to get you to do with the diffamp, you need a bipolar input to the amp. Normal opamp amps with a single input use the chip ground reference are not good for this.
With the tiny 50mV drop you have for full scale (can't you used a bigger shunt r to get more volts?), you made this a tough problem.