We present a suite of high-resolution hydrodynamic simulations of binaries immersed in circumbinary accretion discs (CBDs). For the first time, we investigate the preferential accretion rate as a function of both eccentricity e(b) and mass ratio q(b) in a densely sampled parameter space, finding that when compared with circular binaries, (i) mass ratios grow more efficiently in binaries on moderately eccentric orbits (0.0 less than or similar to e(b) less than or similar to 0.4), and (ii) high eccentricities (e(b) greater than or similar to 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 time-scale, a potentially observable signature in accreting binaries with short orbital periods. We provide CBD locking angles and precession rates as a function of e(b) and q(b) for our simulation suite.