INA series current sense amplifiers like you're already using are actually great. I'm using the INA226 in several designs. In one of them, using a 3mOhm shunt, I can measure up to a bit above 27A, in sub-1mA increments. With basic calibration, its readings are at least as good as the ones from a B&K Precision 2831E while also being able to read a higher current range. It also measures bus voltage in 1.25mV increments which is also quite precise for its range (0-36V). The voltage readings are spot-on too.
If your readings aren't good, then the problem is most likely your design (and possibly fabrication too). There's many ways to go wrong: bad shunt choice, poor footprint (not kelvin sense), ground offset between the INA and the ADC reference, noise (many possible sources here), poor tolerances, poor placement and layout, classic ADC issues... You name it. Expensive parts or switching vendors won't fix any of this.
If you know what you're doing (reading the datasheet thoroughly helps a lot), that you pick the right shunt, have the right layout and connections, optimal ADC settings for the application (conversion time, averaging, etc), decent software (things like filtering and over-sampling could further help) and basic calibration then the results should be pretty fantastic.