What's the voltage gain of your circuit? My "gut feeling" is that gains of 1000 or more are "difficult". And DC is more difficult than AC, since with AC it's easy to filter-out offset & drift.
when I substitute a very low Vio opamp (OP07A, Vio = 10uV) for the original LM358 I used, the output errors are more like 1 volt!
In a high gain circuit, that might be "normal". How would you characterize that error? If it's a constant DC offset (bias), you should be able to calibrate it out (in hardware or software).
There are 4 or 5 different types of error in a measurement like this:
1. Offset - This is a constant-voltage or constant-count error. For example when you are measuring volts, you might always read 1 Volt high. It is normally calibrated-out by "zeroing" (adjusting your system so that zero reads zero).
2. Gain - This is percentage-of-reading error. For example, your reading may be a constant 10% high. This is normally calibrated-out by adjusting the gain so that the maximum-range reads correctly (after zeroing).
(Offest and gain adjustments are standard in just about every measurement calibration.)
3. Nonlinearity - Once the zero & maximum are reading correctly, intermediate readings may be high or low. With software, you can make one or more different gain adjustments to different sections along the range. But, this kind of calibration-adjustment is not that common. It's usually not needed and it's only used when high-precision is required. And, it's only useful when noise & drift are lower than the non-linearity. Where I work, we have a digital-to-analog converter that's calibrated at 100 or more points along the slope. (This can get tricky, because there can be discontinuities where the gain suddenly changes.)
4. Noise - Random variations. Sometimes you can reduce the noise pickup/input, sometimes you have to build a lower-noise circuit (if possible).
5. Drift - Slower variations or lower-frequency noise. The system may read high for awhile, and then start reading low. Again, you have to build a better circuit (if possible).