Following the advice of a couple of mentors, I have decided to redesign the power section of my ADC module. The LM78L15 and LM79L15 used to convert ±18 V into ±15 V are too noisy, and using a linear regulator to go from 15 V down to 5 V would waste two thirds of the input power, while generating significant amounts of heat, which would induce additional noise.
A better solution would be to feed something like ±18 V and 6 V to the board, then use better linear regulators to get ±15 V, and keep the ADM7150 for going from 6 V to 5 V. Also, it would be best to get the 3.3 V that we need for the digital part of the ADC converter from the converted 5 V level, instead of getting it from the 6 V input.
To get the ±18 V and 6 V inputs, one of my mentors suggested that we use a regulator with high switching frequency (2 MHz) to filter out any remaining switching noise. And multiple such regulators could be used with frequency lock with a phase shift between them, delivering very low EMI emissions, and a virtually ripple free power source. This part will be done on the backplane or PSU board.