Botnets typically utilize domain names generated by Domain Generation Algorithms (DGAs) to establish communication between infected devices and their command-and-control (C&C) servers. Numerous DGA domain detection methods have been proposed to identify botnets promptly. Among these, character-level DGA classifiers based on deep learning have been widely adopted due to their simplicity, efficiency, and capability for real-time monitoring. However, deep learning models are vulnerable to adversarial sample attacks. Researchers have proposed various adversarial DGAs to evade detection by character-level DGA classifiers. Nonetheless, domain names generated by these DGAs still exhibit relatively obvious differences compared to real-world domain names, particularly in the distribution of word-level elements. This paper proposes a new adversarial DGA, called SADGA, based on a Self-Attention Generative Adversarial Network (SAGAN). The training network of SADGA incorporates a self-attention mechanism into the traditional WGAN framework to enhance its generative capabilities further. Additionally, SADGA incorporates word-level elements during training, making the word distribution of the generated domain names more similar to real domain names. This paper trains and evaluates five classic deep learning-based DGA classifiers and selects the best-performing models, ATT-CNN-BiLSTM and MIT classifiers, to assess SADGA alongside other adversarial samples. The experimental results demonstrate that compared with the best PKDGA, the AUC of SADGA is reduced by 5.5% and 4.0%, and the F1 is reduced by 1.5% and 2.6%, demonstrating better anti-detection capabilities. Domain names generated by SADGA that contain word-level elements account for 52%, which is close to the 51% proportion of normal Alexa domain names. This study shows that current deep learning-based DGA classifiers are relatively vulnerable and susceptible to attacks from adversarial DGA samples.