The diagnosis of cervical cancer involves manual screening from cytological images, which is time-intensive and susceptible to human error. So, there has been an extensive research in this area to develop an automated deep learning-based screening tool for efficient and reliable segmentation of the cytoplasm and nucleus from cervical cytology images. In this study, we propose a tailored deep segmentation network i.e. CerviSegNet that combines a transformer-based encoder with a convolutional decoder, built upon a modified U-Net backbone enhanced with attention mechanism, enabling both global context understanding and fine-grained localization. An adaptive, learnable weighted loss function integrated with deep supervision is proposed to improve performance with better gradient flow and accelerate convergence during model training. Evaluated on the open-source Cx22 dataset, CerviSegNet achieved Dice scores of 0.7991 (nucleus) and 0.9491 (cytoplasm) in mixed overlapping/non-overlapping conditions, and 0.7918 (nucleus) and 0.9630 (cytoplasm) in strictly overlapping cases. The segmentation results reported are clear evidence of the architectural novelty with generalization capability, and efficiency of the model in overlapping scenarios.