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We present new estimates of the luminosity function (LF) and star formation rate density (SFRD) for an H$\alpha$ selected sample at $z\sim0.62$ from the Deep And Wide Narrow-band (DAWN) survey. Our results are based on a new H$\alpha$ sample in the extended COSMOS region (compared to Coughlin et al. 2018) with the inclusion of flanking fields, resulting in a total area coverage of $\sim$1.5 deg$^2$. A total of 241 H$\alpha$ emitters were selected based on robust selection criteria using spectro-photometric redshifts and broadband color-color classification. We explore the effect of different dust correction prescriptions by calculating the LF and SFRD using a constant dust extinction correction, A{$_{\textrm{H}\alpha}=1$} mag, a luminosity-dependent correction, and a stellar-mass dependent correction. The resulting H$\alpha$ LFs are well fitted using Schechter functions with best-fit parameters: L$^*=10^{42.24}$ erg s$^{-1}$, $\phi^*=10^{-2.85}$ Mpc$^{-3}$, $\alpha = -1.62$ for constant dust correction, L$^*=10^{42.31}$ erg s$^{-1}$, $\phi^*=10^{-2.8}$ Mpc$^{-3}$, $\alpha=-1.39$ for luminosity-dependent dust correction, and L$^*=10^{42.36}$ erg s$^{-1}$, $\phi^*=10^{-2.91}$ Mpc$^{-3}$, $\alpha = -1.48$, for stellar mass-dependent dust correction. The deep and wide nature of the DAWN survey effectively samples H$\alpha$ emitters over a wide range of luminosities, thereby providing better constraints on both the faint and bright end of the LF. Also, the SFRD estimates $\rho_{\textrm{SFR}}=10^{-1.39}$ M$_{\odot}$yr$^{-1}$Mpc$^{-3}$ (constant dust correction), $\rho_{\textrm{SFR}}=10^{-1.47}$ M$_{\odot}$yr$^{-1}$Mpc$^{-3}$ (luminosity-dependent dust correction), and $\rho_{\textrm{SFR}}=10^{-1.49}$ M$_{\odot}$yr$^{-1}$Mpc$^{-3}$ (stellar mass-dependent dust correction) are in good agreement with the evolution of SFRD across redshifts ($0 < z < 2$) seen from previous H$\alpha$ surveys., Comment: 16 pages, 8 figures, Accepted for publication in ApJ