Robotic navigation in dense, cluttered environments such as agricultural canopies presents significant challenges due to physical and visual occlusion caused by leaves and branches. Traditional vision-based or model-dependent approaches often fail in these settings, where physical interaction without damaging foliage and branches is necessary to reach a target. We present a novel reactive controller that enables safe navigation for a robotic arm in a contact-rich, cluttered, deformable environment using end-effector position and real-time tactile feedback. Our proposed framework's interaction strategy is based on a trade-off between minimizing disturbance by maneuvering around obstacles and pushing through them to move towards the target. We show that over 35 trials in 3 experimental plant setups with an occluded target, the proposed controller successfully reached the target in all trials without breaking any branch and outperformed the state-of-the-art model-free controller in robustness and adaptability. This work lays the foundation for safe, adaptive interaction in cluttered, contact-rich deformable environments, enabling future agricultural tasks such as pruning and harvesting in plant canopies.