DNA sequence design aims to design a set of high-quality DNA molecules, which need to satisfy the thermodynamic constraint, similarity constraint, H-Measure constraint and other conflicting objective functions. Therefore DNA sequence design is a typical multi-objective optimization problem. High-quality DNA molecules can effectively prevent non-specific hybridization, undesired secondary structures, and unstable chemical properties during DNA computing, ensuring the reliability and validity of DNA computing. The main deficiency of the existing algorithms is that they may frequently fall into the local optimum. To address this problem, a dual-sorting based constrained multi-objective evolutionary algorithm (DS-CMOEA) is proposed in this paper. DS-CMOEA presents an evaluation index Balance to guide the algorithm to select the solution with similarity and H-Measure equilibrium, and proposes a new fitness function to dynamically adjust the direction of population evolution to avoid the algorithm getting trapped in local optimum. The experimental results indicate that DS-CMOEA exhibits strong global search capability and produces high-quality DNA sequences that outperform those generated by other comparative algorithms.