Texas Spectroscopic Search for Ly$\alpha$ Emission at the End of
  Reionization III. the Ly$\alpha$ Equivalent-width Distribution and Ionized
  Structures at $z > 7$
  Ly$\alpha$ emission from galaxies can be utilized to characterize the
ionization state in the intergalactic medium (IGM). We report our search for
Ly$\alpha$ emission at $z>7$ using a comprehensive Keck/MOSFIRE near-infrared
spectroscopic dataset, as part of the Texas Spectroscopic Search for Ly$\alpha$
Emission at the End of Reionization Survey. We analyze data from 10 nights of
MOSFIRE observations which together target 72 high-$z$ candidate galaxies in
the GOODS-N field, all with deep exposure times of 4.5-19 hr. Utilizing an
improved automated emission-line search, we report 10 Ly$\alpha$ emission lines
detected ($>$4$\sigma$) at $z>7$, significantly increasing the
spectroscopically confirmed sample. Our sample includes large equivalent-width
(EW) Ly$\alpha$ emitters ($>$50\r{A}), and additional tentative Ly$\alpha$
emission lines detected at 3 - 4$\sigma$ from five additional galaxies. We
constrain the Ly$\alpha$ EW distribution at $z\sim7.6$, finding a significant
drop from $z\lesssim6$, suggesting an increasing fraction of neutral hydrogen
(HI) in the IGM in this epoch. We estimate the Ly$\alpha$ transmission through
the IGM ($=$EW$_{z\sim\text{7.6}}$/EW$_{z\sim\text{2-6}}$), and infer an IGM HI
fraction ($X_{\text{HI}}$) of $49^{+19}_{-19}\%$ at $z\sim7.6$, which is lower
in modest tension ($>$1$\sigma$) with recent measurements at $z \sim$ 7.6. The
spatial distribution of the detected Ly$\alpha$ emitters implies the presence
of a potential highly ionized region at $z\sim7.55$ which hosts four Ly$\alpha$
emitters within a $\sim$ 40 cMpc spatial separation. The prominence of this
ionized region in our dataset could explain our lower inferred value of
$X_{\text{HI}}$, though our analysis is also sensitive to the chosen reference
Ly$\alpha$ EW distribution values and reionization models.
