While the ultimate goal of solving computationally intractable problems is to find provably optimal solutions, practical constraints of real-world scenarios often necessitate focusing on efficiently obtaining high-quality, near-optimal solutions. The Quantum Approximate Optimization Algorithm (QAOA) is a state-of-the-art hybrid quantum-classical approach for tackling these challenging problems that are encoded using quadratic and higher-order unconstrained binary optimization problems (QUBO and HUBO). We present SCOOP, a novel QAOA-based framework for solving constrained optimization problems. SCOOP transforms a constrained problem into an unconstrained counterpart, forming SCOOP problem twins. The QAOA quantum algorithm operates on the unconstrained twin to identify potential optimal and near-optimal solutions followed by classical post-processing to obtain the solutions to the constrained twin. We demonstrate our approach on a selection of problems that can be encoded as QUBOs (such as Minimum Vertex Cover and Maximum Independent Set) and HUBOs (such as Minimum Dominating Set and Minimum Maximal Matching).