Forestry cranes operate in dynamic, unstructured outdoor environments where simultaneous collision avoidance and payload sway control are critical for safe navigation. Existing approaches address these challenges separately, either focusing on sway damping with predefined collision-free paths or performing collision avoidance only at the global planning level. We present the first collision-free, sway-damping model predictive controller (MPC) for a forestry crane that unifies both objectives in a single control framework. Our approach integrates LiDAR-based environment mapping directly into the MPC using online Euclidean distance fields (EDF), enabling real-time environmental adaptation. The controller simultaneously enforces collision constraints while damping payload sway, allowing it to (i) replan upon quasi-static environmental changes, (ii) maintain collision-free operation under disturbances, and (iii) provide safe stopping when no bypass exists. Experimental validation on a real forestry crane demonstrates effective sway damping and successful obstacle avoidance. A video can be found at https://youtu.be/tEXDoeLLTxA.