Edit / Oops.
I posted this in the wrong thread. I'm tempted to delete it, but leave it because Kleinstein gave a response and deleting it now would break that answer.
The intended thread was:
https://www.eevblog.com/forum/projects/making-an-lvdt/
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.
I guess both methods are sensitive to nearby ferromagnetic materials which influence the coupling or inductance of the coil(s). But LVDT is also sensitive to the amplitude of the input signal, but it may be less sensitive to humidity or temperature. It would be interesting to see with which method you can get the most resolution and stability.