Hi Ciobanu,
The analog and digital grounds of the ADS1258 need to be at the same voltage potential. Internal to the ADC, the analog and digital circuits are NOT isolated. Isolating them externally will cause problems like you are seeing.
So you may ask, "Why are there separate analog and digital supplies?".
The reason is to partition the analog and digital return currents. (Notice I said "partition" instead of "isolate"). This helps prevent digital noise from coupling into the sensitive analog circuitry. It also allows you to set the digital logic level in some cases.
It sounds like you have ground noise coupling into the analog circuitry. Typically, the PCB layout is the most common culprit when this happens. We usually recommend using a single solid ground plane for both the analog and digital circuitry, but partitioning the board such that you don't have digital return currents flowing in the analog portion.
Regards,
Chris