  We present a suite of high resolution hydrodynamic simulations of binaries
immersed in circumbinary accretion disks (CBDs). For the first time, we
investigate the preferential accretion rate as a function of both eccentricity
$e_{\rm b}$ and mass ratio $q_{\rm b}$ in a densely sampled parameter space,
finding that when compared with circular binaries, mass ratios grow more
efficiently in binaries on moderately eccentric orbits ($0.0 \lesssim e_{\rm b}
\lesssim 0.4$), and high eccentricities ($e_{\rm b} \gtrsim 0.6 $) suppress
mass ratio growth. We suggest that this non-monotonic preferential accretion
behaviour may produce an observable shift in the mass ratio distributions of
stellar binaries and massive black hole binaries. We further find that the
response of a CBD can be divided into three regimes, depending on eccentricity
and mass ratio: (i) CBDs around circular binaries always precess freely,
whereas CBDs around eccentric binaries either (ii) undergo forced precession or
(iii) remain locked at an angle with respect to the binary periapsis. Forced
precession in eccentric binaries is associated with strong modulation of
individual accretion rates on the precession timescale, a potentially
observable signature in accreting binaries with short orbital periods. We
provide CBD locking angles and precession rates as a function of $e_{\rm b}$
and $q_{\rm b}$ for our simulation suite.
