I've been reading a bit about LVDT's and not really convinced this is a wonderful thing. It uses a complicated way to generate
an analog voltage, and then you have to put that into an ADC.
I also bumped into the AD598. This little chip costs EUR122 from mouser, which seems quite ridiculous.
Now you've built your sensor coils and are doing some experiments, I suggest you also do some experiments with a simpler method.
Ever since the '90-ies there are a bunch of circuits floating around the 'net for an LC meter. It's build as a free running LC oscillator and a microcontroller to measure the frequency of the oscillator though some reciprocal math. Years ago I did some experiments with this, and you can do quick measurements with about 0.1Hz resolution, (and could get more resolution by integrating over more oscillation periods). The hardware is very simple, and schematics and software is floating around the 'net.
Note:
Now re-posted in the right thread. I accidentally posted this in:
https://www.eevblog.com/forum/projects/diy-variable-ac-voltage-and-frequency-source-for-calibration/msg5553581