I don't think you will be able to easily overcome this problem: on one hand you have 16-bit ADC (≈75µV resolution) switched by a relay contact, on the other hand there is the 75mA swithing current ..
On option will be to replace relays with CMOS switches, for example MAX322, CD4066, or others, under condition, that you can re-design PCBs to accomodate these changes ..
If you can't go this path, and I assume that you have diodes in parallel with RLYs winding, bypass caps and all of that stuff, then I would suggest that yo re-arange the way you read the ADCs: apply software delay(s) after you switch relay(s) and before you start AD conversion ..
Regards,
IanP