  We present Keck Cosmic Web Imager (KCWI) Ly$\alpha$ integral field
spectroscopy of the fields surrounding 14 Damped Ly$\alpha$ absorbers (DLAs) at
$z \approx 2$. Of these 14 DLAs, 9 have high metallicities ([M/H]$~> -0.3$),
and 4 of those 9 feature a CO-emitting galaxy at an impact parameter $\lesssim
30$ kpc. Our search reaches median Ly$\alpha$ line flux sensitivities of $\sim
2 \times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ over apertures of $\sim6$ kpc and out
to impact parameters of $\sim50$ kpc. We recover the Ly$\alpha$ flux of three
known Ly$\alpha$-emitting H I-selected galaxies in our sample. In addition, we
find two Ly$\alpha$ emitters at impact parameters of $\approx 50-70$ kpc from
the high metallicity DLA at $z \approx 1.96$ toward QSO B0551-366. This field
also contains a massive CO-emitting galaxy at an impact parameter of $\approx
15$ kpc. Apart from the field with QSO B0551-366, we do not detect significant
Ly$\alpha$ emission in any of the remaining 8 high-metallicity DLA fields.
Considering the depth of our observations and our ability to recover previously
known Ly$\alpha$ emitters, we conclude that H I-selected galaxies associated
with high-metallicity DLAs at $z \approx 2$ are dusty, and therefore might
feature low Ly$\alpha$ escape fractions. Our results indicate that
complementary approaches -- using Ly$\alpha$, CO, H$\alpha$, and [C II]
158$\mu$m emission -- are necessary to identify the wide range of galaxy types
associated with $z \approx 2$ DLAs.
