This paper presents a randomly overlapping dual-component radar signals recognition method based on a convolutional neural network-swin transformer (CNN-ST) under different signal-to-noise ratio (SNR), for improving the lower recognition performance and the higher computational costs of the conventional methods. To enhance the feature representation ability and decrease the loss of the detailed features of dual-component radar signals under different SNR, the swin transformer is adopted and integrated into the designed CNN model. An inverted residual structure and lightweight depthwise convolutions are used to maintain the powerful representational ability. The results show that the dual-component radar signals recognition accuracy of the proposed CNN-ST is up to 82.58% at -8 dB, which shows the better recognition performance of the CNN-ST over others. The dual-component radar signals recognition accuracies under different SNR are all more than 88%, which verified that the CNN-ST achieves better recognition accuracy under different SNR. The recognition performance of 2FSK-EQFM, 2FSK-LFM, 2FSK-NS, EQFM-LFM, EQFM-NS, and LFM-NS are up to 94.44%, 93.33%, 88.89%, 95.56%, 87.78%, and 94.44% at SNR of -8 dB, respectively. This work offers essential guidance in enhancing dual-component radar signals recognition under different SNR and promoting actual applications.