Preprint. This framework combines discrete-gradient techniques with scalar auxiliary variables to build numerical integrators that retain important energy properties of conservative and dissipative systems. An extended-state formulation leads to three schemes with different accuracy and computational costs: a first-order semi-implicit method and two second-order alternatives. The construction also covers almost-Poisson systems and conditions for preserving selected Casimir invariants. Tests include phase-field equations, a Hamiltonian oscillator and a Poisson system.