The rapid evolution of Large Language Models (LLMs) has shifted focus from general-purpose capabilities to domain-specific expertise. However, adapting LLMs to specialized fields such as medicine presents two challenge: (1) the "Stability-Plasticity Dilemma", where the model must acquire complex clinical knowledge without suffering from catastrophic forgetting of general world knowledge; and (2) "Task Interference", where disparate sub-tasks, such as medical diagnosis, report summarization, and drug-drug interaction prediction, compete for limited low-rank parameter space. In this paper, we propose Med-MoE-LoRA, a novel framework that integrates Mixture-of-Experts (MoE) with Low-Rank Adaptation (LoRA) to enable efficient multi-task domain adaptation, especially for medical scenarios. Drawing inspiration from recent advances, our framework employs an asymmetric expert distribution where deeper layers are equipped with a higher density of LoRA experts to capture complex semantic abstractions. We further introduce a "Knowledge-Preservation Plugin", inspired by LoRA MoE, to isolate and protect general-purpose reasoning. By utilizing soft merging with adaptive routing and rank-wise decoupling, Med-MoE-LoRA achieves superior performance in medical benchmarks while reducing interference. Experimental results demonstrate that our approach consistently outperforms standard LoRA and conventional MoE architectures across multiple clinical NLP tasks while retaining the model's general cognitive capabilities.