Is there any relation of ADC's conversion rate with its current/power requirement?
Step one is getting rid of your 1N4148 diodes to "make 3.3V", because that's not even likely going to be 3.3V.

And look at your datasheets:

Your Oscillator needs 3.3V +/-5%. The 3.5V to 3.7V made by those diodes doesn't fit the bill, so could one be working and another not? Possibly.

But much more importantly, you say that it is your "3.3V_AVDD", I do not see you convert that to the actual net AVDD, so you may have paid attention to it, but it worries me none the less. Did you look at page 4 of your AD converter?

AVDD: min: 1.7V; max: 1.9V
DRVDD: min 1.7V; max: 3.6V (again the diodes? will they? won't they?)

Are you taking care of that with an actual regulator?

If not, be aware you may just be in the process of just destroying your ADC and this could be the first step in its downfall.

Things with this level of internal complexity aren't joking around when they say they have a maximum voltage to think about.