Solution
The PXIe-4340 User Manual states that the demodulation process uses delay filters to extract the relative amplitudes and phases of the voltages on the primary and secondary LVDT windings. Therefore, small amplitude signals can cause the phase detector to become noisy, causing the signal jump. Additionally, it’s common to see LVDTs that have a specification labelled as "residual offset at null" (meaning, as they cross zero). In this context, the signal from the secondary coil never goes all the way to zero. Instead, it has a period of time where it starts to shift its phase. Once the phase shifts enough, the phase detection circuitry of the PXIe-4340 interprets the phase as flipped, and then you will see the jump to the other side at the current amplitude of the secondary coil signal. Even though the PXIe-4340 has measurement sensitivity in the µV/V range, its low noise plus relatively high bandwidth measurement capabilities result in it seeing the impacts of noise and/or residual offset at null from the LVDT.
In summary, this is an expected behavior based on the theory of operation of LVDTs and the residual offset at null specification of your sensor. You can reduce this by using LVDTs with better residual offset and noise performance, although we can't unfortunately provide any recommendation on a specific model of LVDT.