Robots often face manipulation tasks in environments where vision is inadequate due to clutter, occlusions, or poor lighting--for example, reaching a shutoff valve at the back of a sink cabinet or locating a light switch above a crowded shelf. In such settings, robots, much like humans, must rely on contact feedback to distinguish free from occupied space and navigate around obstacles. Many of these environments often exhibit strong structural priors--for instance, pipes often span across sink cabinets--that can be exploited to anticipate unseen structure and avoid unnecessary collisions. We present a theoretically complete and empirically efficient framework for manipulation in the blind that integrates contact feedback with structural priors to enable robust operation in unknown environments. The framework comprises three tightly coupled components: (i) a contact detection and localization module that utilizes joint torque sensing with a contact particle filter to detect and localize contacts, (ii) an occupancy estimation module that uses the history of contact observations to build a partial occupancy map of the workspace and extrapolate it into unexplored regions with learned predictors, and (iii) a planning module that accounts for the fact that contact localization estimates and occupancy predictions can be noisy, computing paths that avoid collisions and complete tasks efficiently without eliminating feasible solutions. We evaluate the system in simulation and in the real world on a UR10e manipulator across two domestic tasks--(i) manipulating a valve under a kitchen sink surrounded by pipes and (ii) retrieving a target object from a cluttered shelf. Results show that the framework reliably solves these tasks, achieving up to a 2x reduction in task completion time compared to baselines, with ablations confirming the contribution of each module.