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The large spin-orbit misalignments in the DI Herculis stellar binary system have resolved the decades-long puzzle of the anomalously slow apsidal precession rate, but raise new questions regarding the origin of the obliquities. This paper investigates obliquity evolution in stellar binaries hosting modestly-inclined circumbinary disks. As the disk and binary axes undergo mutual precession, each oblate star experiences a torque from its companion star, so that the spin and orbital axes undergo mutual precession. As the disk loses mass through a combination of winds and accretion, the system may be captured into a high-obliquity Cassini state (a spin-orbit resonance). The final obliquity depends on the details of the disk dispersal. We construct a simple disk model to emulate disk dispersal due to viscous accretion and photoevaporation, and identify the necessary disk properties for producing the observed obliquities in DI Herculis. The disk must be massive (at least $10 \%$ of the binary mass). If accretion onto the binary is suppressed, the observed high stellar obliquities are reproduced with a binary-disk inclination of $\sim 5^\circ - 10^\circ$, but if substantial accretion occurs, the inclination must be larger, $\sim 20^\circ - 30^\circ$. If moderate accretion occurs, initially the disk must lose its mass slowly, but eventually lose its remaining mass abruptly, analogous to the observed two-timescale behavior for disks around T-Tauri stars. The spin feedback on the binary orbit causes the binary-disk inclination to decay as the obliquity evolves, a feature that is absent from the standard Cassini state treatment.