Digital electronics based on von Neumann's architecture is reaching its limits to solve large-scale problems essentially due to the memory fetching. Instead, recent efforts to bring the memory near the computation have enabled highly parallel computations at low energy costs. Oscillatory neural network (ONN) is one example of in-memory analog computing paradigm consisting of coupled oscillating neurons. When implemented in hardware, ONNs naturally perform gradient descent of an energy landscape which makes them particularly suited for solving optimization problems. Although the ONN computational capability and its link with the Ising model are known for decades, implementing a large-scale ONN remains difficult. Beyond the oscillators' variations, there are still design challenges such as having compact, programmable synapses and a modular architecture for solving large problem instances. In this paper, we propose a mixed-signal architecture named Saturated Kuramoto ONN (SKONN) that leverages both analog and digital domains for efficient ONN hardware implementation. SKONN computes in the analog phase domain while propagating the information digitally to facilitate scaling up the ONN size. SKONN's separation between computation and propagation enhances the robustness and enables a feed-forward phase propagation that is showcased for the first time. Moreover, the SKONN architecture leads to unique binarizing dynamics that are particularly suitable for solving NP-hard combinatorial optimization problems such as finding the weighted Max-cut of a graph. We find that SKONN's accuracy is as good as the Goemans-Williamson 0.878-approximation algorithm for Max-cut; whereas SKONN's computation time only grows logarithmically. We report on Weighted Max-cut experiments using a 9-neuron SKONN proof-of-concept on a printed circuit board (PCB). Finally, we present a low-power 16-neuron SKONN integrated circuit and illustrate SKONN's feed-forward ability while computing the XOR function.