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We present an analysis of 12 optically selected dual AGN candidates at $z < 0.34$. Each candidate was originally identified via double-peaked [O III] $λ$5007 emission lines, and have received follow-up $Chandra$ and $HST$ observations. Because the X-ray data are low-count ($<100$ counts) with small separations ($<1$"), a robust analysis is necessary for classifying each source. Pairing long-slit [O III] observations with existing $Chandra$ observations, we re-analyze the X-ray observations with ${\tt BAYMAX}$ to determine whether the X-ray emission from each system is more likely a single or dual point source. We find that 4 of the 12 sources are likely dual X-ray point source systems. We examine each point source's spectra via a Monte Carlo method that probabilistically identifies the likely origin of each photon. When doing so, we find that (i) the secondary X-ray point sources in 2 of the systems have $L_{\mathrm{X}}<10^{40}$ erg s$^{-1}$, such that we cannot rule out a non-AGN origin, (ii) one source has a secondary with $L_{\mathrm{X}}>10^{40}$ erg s$^{-1}$ but a spectrum that is too soft to definitively preclude being X-ray emitting diffuse gas that was photoionized by the primary AGN, and (iii) one system (SDSS J1126+2944) is a dual AGN. Additionally, using complementary $HST$ observations, we analyze a sub-sample of systems that are visually identified as merging. Our results suggest that dual AGNs may preferentially reside in mergers with small separations, consistent with both simulations and observations.