In this work, we derive a posteriori error estimates for a class of doubly nonlinear and degenerate elliptic equations, including the Stefan problem and fast and slow diffusion in porous media. Our approach employs equilibrated flux reconstructions, providing guaranteed and fully computable upper bounds on an energy-type norm and local efficiency. These bounds remain independent of the strength of nonlinearity and degeneracy rates. These estimators drive an adaptive solver, dynamically switching between nonlinear solvers to achieve optimal iterations. The adaptive algorithm accounts for discretization, regularization, quadrature, and linearization error components. When Newton's method encounters challenges in achieving convergence, the adaptive algorithm transitions to the L-scheme solver. This solver optimally precomputes the stabilization (or tuning) parameter L > 0 during an offline phase, mirroring the behavior of the Jacobian. The adaptive algorithm is exemplified through four prototypical examples, showcasing its effective error control and notable computational savings.