The ACS712 (obsolete part, replaced by SMD ACS723) with a 5 V supply rail, produces exactly half the supply voltage on its output pin when zero current is flowing between the input pins. With AC current applied, the output will therefore vary (sinusoidally) about a "zero" output of 2.5 V DC, going more positive than 2.5 V during the positive excursions of the AC wave form, and going less positive than 2.5 V during the negative excursions of the AC wave form.
Since all you want to do is sense small changes in AC current, you need to capacitively couple the output of the ACS712 to eliminate its DC component and rectify the result. Your two-diode, single op-amp circuit will do this nicely. Use a 0.1 μF blocking capacitor and a 1 KΩ input resistor to the op-amp. Adjust the feedback resistor upwards from 10 KΩ to as high as 1 MΩ to vary the sensitivity. I would start at the low end of that range. You may also need to add a low-pass filter RC network (series resistor, shunt capacitor) to the op-amp output to "smooth" the rectified AC signal and obtain a noise-free signal suitable for subsequent digital signal processing.
Retain the original offset nulling circuit to produce a nearly zero output with zero current input. Make sure the adjustment pot is much smaller in value than the two fixed resistors on either end of it. You will only need to vary the offset voltage by a few millivolts at the most. Measure the offset nulling voltage with your multimeter between the wiper arm of the pot and circuit common. When making the offset adjustment, there will be a range of adjustment where the output varies, followed by a range where adjustment produces NO change in output. Start with the pot in a position where a slight rotation does produce a change in output, then continue in the direction that changes the output toward zero. Stop adjusting when zero is reached or the output quits changing, then reverse the pot rotation slightly to verify that the minimum output voltage has been achieved. Return the pot rotation to its previously determined minimum output voltage position.
You may be able to get by without any offset adjustment at all, in which case just connect the non-inverting input of the op-amp to circuit common. Circuit common with your potentially lethal mains power supply circuit, is the common connection of the two zener diodes. DO NOT CONNECT CIRCUIT COMMON TO ANY MAINS WIRES OR TO "EARTH" GROUND.
Where are you getting the 5 V necessary to operate the ACS712? You could use two 5 V zener diodes instead of 7 V zener diodes and operate the ACS712 from the resulting +5 V DC rail. The op-amp should work fine with ±5 V DC rails, although its output swing will be less than it would be if ±7 V DC rails are used.
Why is it taking so long to complete such a simple project?