Efficiently serving large language models (LLMs) under dynamic and bursty workloads remains a key challenge for real-world deployment. Existing serving frameworks and static model compression techniques fail to adapt to workload fluctuations, leading to either service-level objective (SLO) violations under full-precision serving or persistent accuracy degradation with static quantization. To deal with these issues, we present MorphServe, a dynamic, workload-aware LLM serving framework based on morphological adaptation. MorphServe introduces two asynchronous, token-level runtime mechanisms: quantized layer swapping, which selectively replaces less impactful layers with quantized alternatives during high-load periods, and pressure-aware KV cache resizing, which repurposes the freed memory to dynamically expand KV cache capacity. These mechanisms enable state-preserving transitions that jointly coordinate weight precision and KV capacity at runtime. Extensive experiments on Vicuna and Llama family models with real-world workloads demonstrate that MorphServe reduces average SLO violations by 92.45% and improves P95 TTFT by 2.2×–3.9× over full-precision serving, without compromising generation quality. Compared to planning-based quantization methods, MorphServe reduces average accuracy degradation by 41.3%, and lowers P95 TTFT by up to 2.4× over KV cache compression while maintaining higher generation quality. These results establish MorphServe as a practical and elastic solution that effectively navigates the accuracy–efficiency Pareto frontier under dynamic LLM serving workloads.