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Binary black holes (BBHs) embedded within the accretion disks that fuel active galactic nuclei (AGN) are promising progenitors for the source of gravitational wave (GW) events detected by LIGO/VIRGO. Several recent studies have shown that when these binaries form, they are likely to be highly eccentric and retrograde. However, many uncertainties remain concerning the orbital evolution of these binaries as they either inspiral toward merger or disassociate. Previous hydrodynamical simulations exploring their orbital evolution have been predominantly two-dimensional or have been restricted to binaries on nearly circular orbits. We present the first high-resolution, three-dimensional local shearing-box simulations of both prograde and retrograde eccentric BBHs embedded in AGN disks. We find that retrograde binaries shrink several times faster than their prograde counterparts and exhibit significant orbital eccentricity growth, the rate of which monotonically increases with binary eccentricity. Our results suggest that retrograde binaries may experience runaway orbital eccentricity growth, which may bring them close enough together at pericenter for GW emission to drive them to coalescence. Although their eccentricity is damped, prograde binaries shrink much faster than their orbital eccentricity decays, suggesting they should remain modestly eccentric as they contract toward merger. Finally, binary precession driven by the AGN disk may dominate over precession induced by the supermassive black hole depending on the binary accretion rate and its location in the AGN disk, which can subdue the evection resonance and von Ziepel–Lidov–Kozai cycles. [ABSTRACT FROM AUTHOR]