Algorithms based on non-unitary evolution have attracted much interest for ground state preparation on quantum computers. One recently proposed method makes use of ancilla qubits and controlled unitary operators to implement weak measurements related to imaginary-time evolution. In this work we specialize and extend this approach to the setting of combinatorial optimization. We first generalize the algorithm from exact to approximate optimization, taking advantage of several properties unique to classical problems. In particular we show how to select parameters such that the success probability of each measurement step is lower bounded away from 1/2. Our approach is general and may be applied to easy-to-prepare initial states as a standalone algorithm, or deployed as a quantum post-processing stage to enhance the performance of a given (parameterized) quantum circuit. We then show how to adapt our paradigm to the important setting of constrained optimization for different classes of hard problem constraints. For this we compare and contrast both penalty-based and feasibility-preserving approaches, elucidating a significant advantage of the latter: feasible initial states are guaranteed to remain feasible under our weak measurement scheme without the need for additional quantum resources. We then propose a more sophisticated variant of our algorithm that adaptively applies additional eigenstate scrambling operators based on the measurement outcomes obtained so far as to further improve performance. In particular, we show that mixing operators from the quantum alternating operator ansatz can be imported directly, both for the necessary scrambling operators and for initial state preparation. For each algorithm we assess quantum and classical resource trade-offs.