The ability to dynamically extend a model to new data and classes is critical for multiple organ and tumor segmentation. However, due to privacy regulations, accessing previous data and annotations can be problematic in the medical domain. This poses a significant barrier to preserving the high segmentation accuracy of the old classes when learning from new classes because of the catastrophic forgetting problem. In this paper, we first empirically demonstrate that simply using high-quality pseudo labels can fairly mitigate this problem in the setting of organ segmentation. Furthermore, we put forward an innovative architecture designed specifically for continuous organ and tumor segmentation, which incurs minimal computational overhead. Our proposed design involves replacing the conventional output layer with a suite of lightweight, class-specific heads, thereby offering the flexibility to accommodate newly emerging classes. These heads enable independent predictions for newly introduced and previously learned classes, effectively minimizing the impact of new classes on old ones during the course of continual learning. We further propose incorporating Contrastive Language-Image Pretraining (CLIP) embeddings into the organ-specific heads. These embeddings encapsulate the semantic information of each class, informed by extensive image-text co-training. The proposed method is evaluated on both in-house and public abdominal CT datasets under organ and tumor segmentation tasks. Empirical results suggest that the proposed design improves the segmentation performance of a baseline neural network on newly-introduced and previously-learned classes along the learning trajectory.