In neoclassical calculations for stellarators, it is customary to retain poloidal E × B precession in the kinetic equation, but to neglect other electric field terms of the same formal magnitude, causing non-conservation of total energy (kinetic plus potential) and magnetic moment. Here we consider the effects of retaining these terms in several types of optimized stellarators. It is challenging to maintain the full conservation laws in analytical calculations for a finite temperature gradient, but it is possible for low collisionality in the case of perfectly quasisymmetric plasmas, or more generally, for omnigenous plasmas. For the omnigenous calculation, we develop a new ordering that allows an expansion about the quasisymmetric solution. Even though canonical angular momentum is not conserved in an omnigenous nonsymmetric plasma, it is still possible to define an effective canonical angular momentum for a corresponding quasisymmetric system, and this quantity is useful as a radial variable in the analysis of the omnigenous plasma. Applying these observations and techniques to the drift-kinetic equation, electric-field-driven modifications to the omnigenous distribution function are derived, including a new term which has no analogue in previous neoclassical calculations for omnigenous or quasisymmetric stellarators.