The healthcare industry has witnessed significant transformations in e-health services where Electronic Health Records (EHRs) are transferred to mobile edge clouds to facilitate healthcare. Many edge cloud-based system designs have been proposed, but some technical challenges still remain, such as low quality of services (QoS), data privacy and system security due to centralized healthcare architectures. In this paper, we propose a novel hybrid approach of data offloading and data sharing for healthcare using edge cloud and blockchain. First, an efficient data offloading scheme is proposed where IoT health data can be offloaded to nearby edge servers for data processing with privacy awareness. Then, a data sharing scheme is integrated to enable data exchange among healthcare users via blockchain. Particularly, a trustworthy access control mechanism is developed using smart contracts for access authentication to achieve secure EHRs sharing. Implementation results from extensive real-world experiments show the superior advantages of the proposal over the existing schemes in terms of improved QoS, enhanced data privacy and security, and low smart contract costs.