  The evolution of magnetism in late-type dwarfs remains murky, as we can only
weakly predict levels of activity for M dwarfs of a given mass and age. We
report results from our spectroscopic survey of M dwarfs in the Southern
Continuous Viewing Zone (CVZ) of the Transiting Exoplanet Survey Satellite
(TESS). As the TESS CVZs overlap with those of the James Webb Space Telescope,
our targets constitute a legacy sample for studies of nearby M dwarfs. For 122
stars, we obtained at least one $R\approx 2000$ optical spectrum with which we
measure chromospheric $\mathrm{H}\alpha$ emission, a proxy for magnetic field
strength. The fraction of active stars is consistent with what is expected for
field M dwarfs; as in previous studies, we find that late-type M dwarfs remain
active for longer than their early type counterparts. While the TESS light
curves for $\approx$20% of our targets show modulations consistent with
rotation, TESS systematics are not well enough understood for confident
measurements of rotation periods ($P_{\mathrm{rot}}$) longer than half the
length of an observing sector. We report periods for 12 stars for which we
measure $P_{\mathrm{rot}} {\lower0.8ex\hbox{$\buildrel <\over\sim$}}$ 15 d or
find confirmation for the TESS-derived $P_{\mathrm{rot}}$ in the literature.
Our sample of 21 $P_{\mathrm{rot}}$, which includes periods from the
literature, is consistent with our targets being spun-down field stars.
Finally, we examine the $\mathrm{H}\alpha$-to-bolometric luminosity
distribution for our sample. Two stars are rotating fast enough to be
magnetically saturated, but are not, hinting at the possibility that fast
rotators may appear inactive in $\mathrm{H}\alpha$.
