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Open radio access networks (e.g., O-RAN) facilitate fine-grained control (e.g., near-RT RIC) in next-generation networks, necessitating advanced AI/ML techniques in handling online resource orchestration in real-time. However, existing approaches can hardly adapt to time-evolving network dynamics in network slicing, leading to significant online performance degradation. In this paper, we propose AdaSlicing, a new adaptive network slicing system, to online learn to orchestrate virtual resources while efficiently adapting to continual network dynamics. The AdaSlicing system includes a new soft-isolated RAN virtualization framework and a novel AdaOrch algorithm. We design the AdaOrch algorithm by integrating AI/ML techniques (i.e., Bayesian learning agents) and optimization methods (i.e., the ADMM coordinator). We design the soft-isolated RAN virtualization to improve the virtual resource utilization of slices while assuring the isolation among virtual resources at runtime. We implement AdaSlicing on an O-RAN compliant network testbed by using OpenAirInterface RAN, Open5GS Core, and FlexRIC near-RT RIC, with Ettus USRP B210 SDR. With extensive network experiments, we demonstrate that AdaSlicing substantially outperforms state-of-the-art works with 64.2% cost reduction and 45.5% normalized performance improvement, which verifies its high adaptability, scalability, and assurance.