  The TESS mission delivers time-series photometry for millions of stars across
the sky, offering a probe into stellar astrophysics, including rotation, on a
population scale. However, light curve systematics related to the satellite's
13.7-day orbit have prevented stellar rotation searches for periods longer than
13 days, putting the majority of stars beyond reach. Machine learning methods
have the ability to identify systematics and recover robust signals, enabling
us to recover rotation periods up to 35 days for GK dwarfs and 80 days for M
dwarfs. We present a catalog of 7245 rotation periods for cool dwarfs in the
Southern Continuous Viewing Zone, estimated using convolutional neural
networks. We find evidence for structure in the period distribution consistent
with prior Kepler and K2 results, including a gap in 10--20-day cool star
periods thought to arise from a change in stellar spin-down or activity. Using
a combination of spectroscopic and gyrochronologic constraints, we fit stellar
evolution models to estimate masses and ages for stars with rotation periods.
We find strong correlations between the detectability of rotation in TESS and
the effective temperature, age, and metallicity of the stars. Finally, we
investigate the relationships between rotation and newly obtained spot filling
fractions estimated from APOGEE spectra. Field star spot filling fractions are
elevated in the same temperature and period regime where open clusters'
magnetic braking stalls, lending support to an internal shear mechanism that
can produce both phenomena.
