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Gastrointestinal cancer is a leading cause of cancer-related incidence and death, making it crucial to develop novel computer-aided diagnosis systems for early detection and enhanced treatment. Traditional approaches rely on the expertise of gastroenterologists to identify diseases; however, this process is subjective, and interpretation can vary even among expert clinicians. Considering recent advancements in classifying gastrointestinal anomalies and landmarks in endoscopic and video capsule endoscopy images, this study proposes a hybrid model that combines the advantages of Transformers and Convolutional Neural Networks (CNNs) to enhance classification performance. Our model utilizes DenseNet201 as a CNN branch to extract local features and integrates a Swin Transformer branch for global feature understanding, combining both to perform the classification task. For the GastroVision dataset, our proposed model demonstrates excellent performance with Precision, Recall, F1 score, Accuracy, and Matthews Correlation Coefficient (MCC) of 0.8320, 0.8386, 0.8324, 0.8386, and 0.8191, respectively, showcasing its robustness against class imbalance and surpassing other CNNs as well as the Swin Transformer model. Similarly, for the Kvasir-Capsule, a large video capsule endoscopy dataset, our model outperforms all others, achieving overall Precision, Recall, F1 score, Accuracy, and MCC of 0.7007, 0.7239, 0.6900, 0.7239, and 0.3871. Moreover, we generated saliency maps to explain our model's focus areas, demonstrating its reliable decision-making process. The results underscore the potential of our hybrid CNN-Transformer model in aiding the early and accurate detection of gastrointestinal (GI) anomalies.