Just thinking out loud and I'm by no means a digital circuits/theory guy...
I can check the clocking, but since it's a multi-channel converter, I would assume the clock is the same across the ADC chips. There would logically be some sort of clock distribution, to all of the ADC chips, but the base clock would be the same for all. Since the problem only occurs with two of the ADC chips, and only one side of one of those chips, I would assume the clock is fine, but something specific to those two chips is at fault.
If the digital clock frequency were unstable, when the input is a pure sine wave the observed signal (as reconverted through DAC) would be a sine with a wavering pitch similar to wow/flutter on an analog deck. Perhaps more specifically, that would be the case with a varying clock. A noisy clock would potentially generate random pops/clicks.
The noise that's experienced is like digital clicking (hash) that only occurs with certain input levels to the ADC. If you change the physical temperature of the ADC chip ( like with a cold spray), you can shift the input level that the digital hash occurs at. A sine wave input at say -20dbFS is perfectly fine, yet the same sine wave at -21dbFS has hash and the hash goes away at -22dbFS.
My gut says ADC chip. I'm open to input though. I've got a 30Mhz scope, I just don't know if I can find an "easy" clock point to stick the probe.
The converter in question is a MOTU HD192. It uses six AK5394VS chips. The two that I'm having trouble with are in the middle of the board (inputs 5, 7 and 8).