  Lyman-Alpha Emitting galaxies (LAEs) are typically young, low-mass,
star-forming galaxies with little extinction from interstellar dust. Their low
dust attenuation allows their Ly$\alpha$ emission to shine brightly in
spectroscopic and photometric observations, providing an observational window
into the high-redshift universe. Narrowband surveys reveal large, uniform
samples of LAEs at specific redshifts that probe large scale structure and the
temporal evolution of galaxy properties. The One-hundred-deg$^2$ DECam Imaging
in Narrowbands (ODIN) utilizes three custom-made narrowband filters on the Dark
Energy Camera (DECam) to discover LAEs at three equally spaced periods in
cosmological history. In this paper, we introduce the hybrid-weighted
double-broadband continuum estimation technique, which yields improved
estimation of Ly$\alpha$ equivalent widths. Using this method, we discover
6339, 6056, and 4225 LAE candidates at $z =$ 2.4, 3.1, and 4.5 in the extended
COSMOS field ($\sim$9 deg$^2$). We find that [O II] emitters are a minimal
contaminant in our LAE samples, but that interloping Green Pea-like [O III]
emitters are important for our redshift 4.5 sample. We introduce an innovative
method for identifying [O II] and [O III] emitters via a combination of
narrowband excess and galaxy colors, enabling their study as separate classes
of objects. We present scaled median stacked SEDs for each galaxy sample,
revealing the overall success of our selection methods. We also calculate
rest-frame Ly$\alpha$ equivalent widths for our LAE samples and find that the
EW distributions are best fit by exponential functions with scale lengths of
$w_0$ = 55 $\pm$ 1, 65 $\pm$ 1, and 62 $\pm$ 1 Angstroms, respectively.
