Yeah, they suck.
As you've found, the current measurement below about 500mA is horrible. The exact reason escapes me (dang brain)... but it is a common issue with "low-side" resistor current sensing - basically, at low currents, the opamp's inputs and output are all ground or very near ground. And the mfg was too stingy to create a negative supply rail, so the opamp's negative power pin is ground too. And the opamp is pole-axed by it's finite common mode input range.
Once the current rises, the sense resistor voltage rises, and the opamp inputs and output rise away from the ground pin.... and all is good.