Safe motion planning for autonomous driving systems in uncertain environments remains a significant challenge. The inherent traffic uncertainties caused by both the uncertain control and behavioral intentions of dynamic surrounding vehicles (SVs) are difficult to anticipate, hindering accurate motion prediction and complicating safe decision-making for the autonomous ego vehicle (EV). To mitigate these challenges, this paper proposes an efficient and safe motion-planning strategy that integrates intention awareness into the uncertainty prediction of SVs. The uncertainty prediction is performed analytically online through forward reachability analysis and is further improved by capturing both the control intentions and behavioral intentions of SVs to support the motion-planning process. The effectiveness of the method is demonstrated in various challenging scenarios, including planning tasks in encounter scenarios, multi-vehicle distributed planning problems, and a multi-vehicle case study based on a recorded real-world traffic dataset.