Training large language models (LLMs) is often constrained by GPU memory limitations. To alleviate memory pressure, activation recomputation and data compression have been proposed as two major strategies. However, both approaches have limitations: recomputation introduces significant training overhead, while compression can lead to accuracy degradation and computational inefficiency when applied naively. In this paper, we propose Adacc, the first adaptive memory optimization framework that unifies activation recomputation and data compression to improve training efficiency for LLMs while preserving model accuracy. Unlike existing methods that apply static, rule-based strategies or rely solely on one technique, Adacc makes fine-grained, tensor-level decisions, dynamically selecting between recomputation, retention, and compression based on tensor characteristics and runtime hardware constraints. Adacc tackles three key challenges: (1) it introduces layer-specific compression algorithms that mitigate accuracy loss by accounting for outliers in LLM activations; (2) it employs a MILP-based scheduling policy to globally optimize memory strategies across layers; and (3) it integrates an adaptive policy evolution mechanism to update strategies during training in response to changing data distributions. Experimental results show that Adacc improves training throughput by 1.01x to 1.37x compared to state-of-the-art frameworks, while maintaining accuracy comparable to the baseline.