The merger of two supermassive black holes (SMBHs) imparts a gravitational-wave (GW) recoil kick to the remnant SMBH, which can even eject the SMBH from its host galaxy. An actively-accreting, recoiling SMBH may be observable as an offset quasar. Prior to the advent of a space-based GW observatory, detections of these offset quasars may offer the best chance for identifying recent SMBH mergers. Indeed, observational searches for recoiling quasars have already identified several promising candidates. However, systematic searches for recoils are currently hampered by large uncertainties regarding how often offset quasars should be observable and where they are most likely to be found. Motivated by this, we have developed a model for recoiling quasars in a cosmological framework, utilizing information about the progenitor galaxies from the Illustris cosmological hydrodynamic simulations. For the first time, we model the effects of BH spin alignment and recoil dynamics based on the gas-richness of host galaxies. We predict that if BH spins are not highly aligned, seeing-limited observations could resolve offset AGN, making them promising targets for all-sky surveys. The rarity of large broad-line offsets among SDSS quasars is likely due in part to selection effects but suggests that spin alignment plays a role in suppressing recoils. Nonetheless, in our most physically motivated model where alignment occurs only in gas-rich mergers, hundreds of offset AGN should be found in all-sky surveys. Our findings strongly motivate a dedicated search for recoiling AGN.