  Two recently discovered white dwarfs, WDJ041246.84$+$754942.26 and
WDJ165335.21$-$100116.33, exhibit H$\alpha$ and H$\beta$ Balmer line emission
similar to stars in the emerging DAHe class, yet intriguingly have not been
found to have detectable magnetic fields. These white dwarfs are assigned the
spectral type DAe. We present detailed follow-up of the two known DAe stars
using new time-domain spectroscopic observations and analysis of the latest
photometric time-series data from TESS and ZTF. We measure the upper magnetic
field strength limit of both stars as $B < 0.05$ MG. The DAe white dwarfs
exhibit photometric and spectroscopic variability, where in the case of
WDJ041246.84$+$754942.26 the strength of the H$\alpha$ and H$\beta$ emission
cores varies in anti-phase with its photometric variability over the spin
period, which is the same phase relationship seen in DAHe stars. The DAe white
dwarfs closely cluster in one region of the Gaia Hertzsprung-Russell diagram
together with the DAHe stars. We discuss current theories on non-magnetic and
magnetic mechanisms which could explain the characteristics observed in DAe
white dwarfs, but additional data are required to unambiguously determine the
origin of these stars.
