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The quest to solve hard combinatorial optimization problems efficiently -- still a longstanding challenge for traditional digital computers -- has inspired the exploration of many alternate computing models and platforms. As a case in point, oscillator networks offer a potentially promising energy efficient and scalable option. However, prior oscillator-based combinatorial optimization solvers have primarily focused on quadratic combinatorial optimization problems that consider only pairwise interaction among the oscillators. In this work, we propose a new computational model based on the maximum entropy production (MEP) principle that exploits higher order interactions among the oscillators, and demonstrate its application in solving the non-quadratic maximum satisfiability (MaxSAT) problem. We demonstrate that the solution to the MaxSAT problem can be directly mapped to the entropy production rate in the oscillator network, and subsequently, propose an area-efficient hardware implementation that leverages Compute-in-Memory (CiM) primitives. Using experiments along with analytical and circuit simulations, we elucidate the performance of the proposed approach in computing high-quality optimal / near-optimal solutions to the MaxSAT problem. Our work not only reveals how oscillators can solve non-quadratic combinatorial optimization problems such as MaxSAT but also extends the application of this dynamical system-based approach to a broader class of problems that can be easily decomposed to the MaxSAT solution.