Directed Acyclic Graph (DAG)-based blockchain (a.k.a distributed ledger) has become prevalent for supporting highly concurrent applications. Its inherent parallel data structure accelerates block generation significantly, shifting the bottleneck from performance to storage scalability. An intuitive solution is to apply state sharding that divides the entire ledger (i.e., transactions and states) into multiple shards. While each node only stores proportional transactions, it suffers from the challenges of storing and ensuring the processing consistency of cross-shard transactions. In this paper, we propose SharDAG, a new mechanism that leverages adaptive sharding for DAG-based blockchains to achieve high performance and strong consistency. The key idea of SharDAG is to exploit unique characteristics - silent assets - and design a lightweight processing mechanism based on avatar account caching. Furthermore, we design a Byzantine resilient cross-shard verification mechanism with a theoretically optimal number of participating nodes, which guarantees the consistency and security of avatar account aggregation. Our comprehensive evaluations on real-world workloads demonstrate that SharDAG presents up to 3.8 x throughput improvement compared to the state-of-the-art and reduces the storage overhead of cross-shard transactions.