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We present a new multi-fluid, multi-temperature plasma solver with adaptive Cartesian mesh (ACM) based on a full-Newton (non-linear, implicit) scheme for collisional low-temperature plasma. The particle transport is described using the drift-diffusion approximation for electrons and ions coupled to Poisson equation for electric field. In addition, the electron-energy transport equation is solved to account for electron thermal conductivity, Joule heating, and energy loss of electrons in collisions with neutral species. The rate of electron-induced ionization is a function of electron temperature and could also depend on electron density (important for plasma stratification). The ion and gas temperature are kept constant. The spatial discretization of the transport equations uses non-isothermal Scharfetter-Gummel scheme from semiconductor physics adapted for multi-dimensional ACM framework. We demonstrate the new solver for simulations of direct current (DC) and radio frequency (RF) discharges. The implicit treatment of the coupled equations allows using large time steps, and the full-Newton method enables fast non-linear convergence at each time step, offering greatly improved efficiency of fluid plasma simulations. We discuss the selection of time steps for solving different plasma problems. The new solver enables us to solve several problems we could not solve before with existing software: two- and three-dimensional structures of the entire DC discharges including cathode and anode regions with electric field reversals, normal cathode spot and anode ring, plasma stratification in diffuse and constricted DC discharges, and standing striations in RF discharges.