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Computationally hard problems, including combinatorial optimization, can be mapped into the problem of finding the ground-state of an Ising Hamiltonian. Building physical systems with collective computational ability and distributed parallel processing capability can accelerate the ground-state search. Here, we present a continuous-time dynamical system (CTDS) approach where the ground-state solution appears as stable points or attractor states of the CTDS. We harness the emergent dynamics of a network of phase-transition nano-oscillators (PTNO) to build an Ising Hamiltonian solver. The hardware fabric comprises of electrically coupled injection-locked stochastic PTNOs with bi-stable phases emulating artificial Ising spins. We demonstrate the ability of the stochastic PTNO-CTDS to progressively find more optimal solution by increasing the strength of the injection-locking signal - akin to performing classical annealing. We demonstrate in silico that the PTNO-CTDS prototype solves a benchmark non-deterministic polynomial time (NP)-hard Max-Cut problem with high probability of success. Using experimentally calibrated numerical simulations and incorporating non-idealities, we investigate the performance of our Ising Hamiltonian solver on dense Max-Cut problems with increasing graph size. We report a high energy-efficiency of 1.3x10^7 solutions/sec/Watt for 100-node dense Max-cut problems which translates to a 5x improvement over the recently demonstrated memristor-based Hopfield network and several orders of magnitude improvement over other candidates such as CPU and GPU, quantum annealer and photonic Ising solver approaches. Such an energy efficient hardware exhibiting high solution-throughput/Watt can find applications in industrial planning and manufacturing, defense and cyber-security, bioinformatics and drug discovery.