DNA sharing and querying of personal genomic sequences have becoming more critical than ever with the accumulation of large-scale biomedical data. Quantitative genomic studies heavily rely on multi-source DNA datasets from different institutions, who are reluctant to share the data via cloud centers. The high sensitivity of DNA has compelled the government to restrict its acquisition and usage. One potential solution to tackle the issue is designing a secure query strategy on the encrypted DNA datasets. However, the popular secure DNA query schemes remain defective in verifiability, reliability, and DNA privacy. To relieve the issues, we propose a DNA searchable encryption method named EncGD to achieve verifiable privacy-preserving queries and reliable updates on multi-source dynamic DNA datasets. Specifically, the proposed scheme EncGD is designed by the plaintext-related permutation and substitution primitives, which can enhance the DNA privacy due to the chosen-plaintext attack (CPA)-resist ability. Furthermore, the method realizes the verifiability and reliability by adding known information to block the malicious behaviors of clouds. Experimental results demonstrate the superior efficiency of the proposed method comparing with the two state-of-the-art schemes in terms of time and space costs.