Before exchanging the reference, I would you just measure the noise of the zener reference - at least of one has a suitable instrument (e.g. noise tester / AC coupled or low noise DMM (may need extra ref. to measure only difference)).
One point would be comparing the noise for reading a short and reading a low noise DC source.
Using the second reference and do the adjustment to the LM399 for every AZ cycle would be strange, as this would add quite some noise - though it would eliminate drift from the integrating resistors. Its a little like doing an simplified ACAL on the 3458 before each reading: it reduces drift, but also add time and thus noise, as less time is available for the actual measurement.
The more logical way would be a slower (e.g. average over 10s of readings) correction of drift from the reference and integrating resistors.
Increasing C821 would make sense if the zener is noisy - so the DW232 should not need it, the old zener might profit from something like 100 µF. The point is reducing noise in the 1-10 kHz range. The switching frequency for the ADC seems to be rather high - so jitter from the reference switching and maybe charge injection could be an issue. Also only switching the positive reference and adding a constant negative parts adds some noise. So the ADC is by design not very low noise.
The AD707 at the LM399 is not an issue: the noise of the LM399 is something like 10 times higher. I would be more concerned about switching noise from the LTC1043 might have an influence on the LM399.