The combination of cloud computing and the Internet of Things has created numerous possibilities for data sharing applications that prioritize privacy, accountability, and efficiency. Currently, many data sharing schemes based on Attribute-Based Encryption (ABE) show promise due to their fine-grained access control and one-to-many data encryption. However, existing schemes often struggle to find a balance between protecting user identity privacy and ensuring accountability. To address this limitation, this paper proposes a Blockchain-Assisted Anonymous Data Sharing scheme (BAADS) that provides full accountability. BAADS incorporates ElGamal encryption, signature of knowledge, and blockchain technology to design an anonymous identity management mechanism. Additionally, it integrates a reconstructed attribute encryption scheme that supports data auditing, identity tracking, and verifiable outsourced decryption. Compared to existing schemes, the key advantage of BAADS is that it enables both anonymous data sharing and full accountability for malicious or negligent behavior of all participants (data owners, data users, and cloud service providers). We have formalized the security aspects of BAADS and demonstrated its security in the random oracle model. Moreover, we have implemented the scheme using Hyperledger Fabric and OpenABE, and the experimental results have shown the effectiveness and availability of the scheme.