  Stellar-mass black holes can become embedded within the gaseous disks of
active galactic nuclei (AGNs). Afterwards, their interactions are mediated by
their gaseous surroundings. In this work, we study the evolution of
stellar-mass binary black holes (BBHs) embedded within AGN disks using a
combination of three-dimensional hydrodynamic simulations and analytic methods,
focusing on environments in which the AGN disk scale height $H$ is $\gtrsim$
the BBH sphere of influence. We model the local surroundings of the embedded
BBHs using a wind tunnel formalism and characterize different accretion regimes
based on the local properties of the disk, which range from wind-dominated to
quasi-spherical. We use our simulations to develop prescriptions for mass
accretion and drag for embedded BBHs. We use these prescriptions, along with
AGN disk models that can represent the Toomre-unstable outer regions of AGN
disks, to study the long-term evolution of the BBHs as they migrate through the
disk. We find that BBHs typically merge within $\lesssim 5-30\,{\rm Myr}$,
increasing their mass significantly in the process, allowing BBHs to enter (or
cross) the pair-instability supernova mass gap. The rate at which gas is
supplied to these BBHs often exceeds the Eddington limit, sometimes by several
orders of magnitude. We conclude that most embedded BBHs will merge before
migrating significantly in the disk. Depending on the conditions of the ambient
gas and the distance to the system, LISA can detect the transition between the
gas-dominated and gravitational wave dominated regime for inspiraling BBHs that
are formed sufficiently close to the AGN ($\lesssim$ 0.1 pc). We also discuss
possible electromagnetic signatures during and following the inspiral, finding
that it is generally unlikely but not inconceivable for the bolometric
luminosity of the BBH to exceed that of the host AGN.
