The multi-objective vehicle routing problem with time windows (MOVRPTW) aims to obtain a Pareto-optimal solution set among multiple conflicting objectives; however, traditional heuristics and exact algorithms often struggle to solve it efficiently. Although multi-objective evolutionary algorithms (MOEAs) have been widely applied, they still exhibit limitations in handling complex constraints and are prone to premature convergence and local optima, which restricts their solution performance. To address these issues, this paper proposes a dual-stage adaptive co-evolutionary algorithm (DS-ACEA). To effectively balance exploration and exploitation, the evolutionary process is divided into two stages. In the first stage, a feasibility-preferable strategy based on constraint violation values (CVs), which follows a grouping-then-selection mechanism, is adopted. This strategy guides the search toward broader feasible regions while improving population diversity. In the second stage, an elitist selection strategy based on Pareto dominance is employed. By preferentially selecting feasible solutions with better objective values, this strategy accelerates the convergence of the population toward the feasible nondominated Pareto front. To evaluate the performance of the proposed algorithm, extensive empirical experiments are conducted on multiple benchmark instances. The results show that DS-ACEA significantly outperforms advanced comparative algorithms on several key performance indicators when solving MOVRPTW, thereby verifying the effectiveness of the constraint-handling technique based on the dual-stage selection strategy.