SANDO is a safe trajectory planner for 3D dynamic unknown environments, where obstacle locations and motions are unknown a priori and a collision-free plan can become unsafe at any moment, requiring fast replanning. Existing soft-constraint planners are fast but cannot guarantee collision-free paths, while hard-constraint methods ensure safety at the cost of longer computation. SANDO addresses this trade-off through three contributions. First, a heat map-based A* global planner steers paths away from high-risk regions using soft costs, and a spatiotemporal safe flight corridor (STSFC) generator produces time-layered polytopes that inflate obstacles only by their worst-case reachable set at each time layer, rather than by the worst case over the entire horizon. Second, trajectory optimization is formulated as a Mixed-Integer Quadratic Program (MIQP) with hard collision-avoidance constraints, and a variable elimination technique reduces the number of decision variables, enabling fast computation. Third, a formal safety analysis establishes collision-free guarantees under explicit velocity-bound and estimation-error assumptions. Ablation studies show that variable elimination yields up to 7.4x speedup in optimization time, and that STSFCs are critical for feasibility in dense dynamic environments. Benchmark simulations against state-of-the-art methods across standardized static benchmarks, obstacle-rich static forests, and dynamic environments show that SANDO consistently achieves the highest success rate with no constraint violations across all difficulty levels; perception-only experiments without ground truth obstacle information confirm robust performance under realistic sensing. Hardware experiments on a UAV with fully onboard planning, perception, and localization demonstrate six safe flights in static environments and ten safe flights among dynamic obstacles.