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 ($>$50r{A}), and additional tentative Ly$alpha$ emission lines detected at 3 - 4$sigma$ from five additional galaxies. We constrain the Ly$alpha$ EW distribution at $zsim7.6$, finding a significant drop from $zlesssim6$, suggesting an increasing fraction of neutral hydrogen (HI) in the IGM in this epoch. We estimate the Ly$alpha$ transmission through the IGM ($=$EW$_{zsimtext{7.6}}$/EW$_{zsimtext{2-6}}$), and infer an IGM HI fraction ($X_{text{HI}}$) of $49^{+19}_{-19}%$ at $zsim7.6$, 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 $zsim7.55$ 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.
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