In genuine nonequilibrium systems that undergo continuous driving, the thermodynamic forces are nonconservative, meaning they cannot be described by any free energy potential. Nonetheless, we show that the dynamics of such systems are governed by a "generalized free energy" that is derived from a large-deviations variational principle. This variational principle also yields a decomposition of fluxes, forces, and dissipation (entropy production) into a conservative "excess" part and a nonconservative "housekeeping" part. Our decomposition is universally applicable to stochastic master equations, deterministic chemical reaction networks, and open systems. We also show that the excess entropy production obeys a thermodynamic speed limit (TSL), a fundamental thermodynamic constraint on the rate of state evolution and/or external fluxes. We demonstrate our approach on several examples, including real-world metabolic networks, where we derive fundamental dissipation bounds and uncover "futile" metabolic cycles. Our generalized free energy and decomposition are empirically accessible to thermodynamic inference in both stochastic and deterministic systems. We discuss important connections to several theoretical frameworks, including information geometry and Onsager theory, as well as previous excess/housekeeping decompositions.