Data storage on synthetic DNA is a novel technique that offers improved durability and density compared to conventional storage supports. Current challenges in the practical implementation of DNA data storage include the high cost of DNA synthesis and the significant amount of errors introduced during DNA sequencing. These errors are not only substitutions but also insertions and deletions, which compromise the sequence synchronization and cannot be corrected using conventional error correction methods. To overcome these issues, this paper proposes an error correction scheme that prioritizes the reduction of DNA synthesis costs through the use of high rate codes. The proposed solution utilizes a consensus algorithm that leverages the inherent redundancy in the output data, followed by a proposed synchronization method for correcting residual insertions and deletions after the consensus. Numerical results show the effectiveness of the proposed method at correcting a large amount of errors, while maintaining a high coding rate.