Implicit time integration methods for nonlinear elastodynamics can exhibit numerical instability in two failure modes: diverging Newton-Raphson iterations or pathological energy growth. Numerical dissipation is commonly employed to stabilize the iterations [1], while energy-conserving integrators preserve long-term energy behavior [2]. Previous energy-decaying integrators [3, 4] attempting to combine these features have suffered from a loss of second-order accuracy or an increase in computational expense with fully implicit multistage designs. In fact, it is impossible to simultaneously achieve second-order accuracy and monotonic energy decay with single-stage (single-step/linear multistep) integrators even for linear problems; at best, bounded oscillatory energy decay can be obtained. We embrace this limitation and relax the monotonicity requirement to develop new families of single-stage implicit integrators for nonlinear elastodynamics that maintain second-order accuracy while exhibiting oscillatory energy decay. The theoretical development is enabled by a novel time-weighted residual approach that decomposes the stress and displacement fields (extending the hybrid strain-displacement approach of Masuri et al [5]). The approach incorporates several distinct second-order extrapolations which obtain identical accuracy and energy properties but alternative implementations. When the numerical dissipation is turned off, the framework recovers the Simo-Gonzalez energy-momentum method (EMM) alongside modified variants with more efficient implementation. Controllable numerical dissipation enhances nonlinear convergence, while oscillatory energy decay preserves long-term stability. We demonstrate accuracy, robustness, and efficiency through representative numerical examples featuring material and geometric nonlinearity. The results show improved nonlinear convergence compared with conventional energy-conserving integrators, without any loss of second-order accuracy or increase in computational cost as with existing energy-decaying integrators. These findings provide a practical, high-fidelity, and computationally efficient framework for long-term simulations in nonlinear elastodynamics.