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Well that second schematic is not actually mine, I inherited it from previous guy. It happens to work with sine wave input, so I didn't modify it.I'm puzzled as to why you have all those caps in parallel/series? Isn't that likely to make the circuit less stable?
If it is offset the AD790 will be better, but you may see multiple pulses at the zero crossing from noise. But certainly worth a try. Is it better with the high amplitude signal?