Thanks for your queries, see responses below your quoted questions:

MZielinski said:

Does sending the register access command and the register data on SDI prevent a normal conversion from occurring?

No, the AD4004 will still perform a sample on CNV rising…

Hi Rob,

I also noticed that the operation supply voltage of the AD4004 is +/- 5V to +/- 15V. Can you confirm that you're providing correct supply voltage and setup for the AD4004? Because if not, the problem might be on the AD4004 and not on the AD8235…

Mohi,

I would suggest taking a look at the AD7685 or the AD4008.  Both have significantly higher maximum throughput than you need but should be able to operate in the range of sample rates you are interested in.  The advantage of the AD4008 is the HI-Z…