Robotic manipulation tasks such as inserting a key into a lock or plugging a USB device into a port can fail when visual perception is insufficient to detect misalignment. In these situations, touch sensing is crucial for the robot to monitor the task's states and make precise, timely adjustments. Current touch sensing solutions are either insensitive to detect subtle changes or demand excessive sensor data. Here, we introduce TranTac, a data-efficient and low-cost tactile sensing and control framework that integrates a single contact-sensitive 6-axis inertial measurement unit within the elastomeric tips of a robotic gripper for completing fine insertion tasks. Our customized sensing system can detect dynamic translational and torsional deformations at the micrometer scale, enabling the tracking of visually imperceptible pose changes of the grasped object. By leveraging transformer-based encoders and diffusion policy, TranTac can imitate human insertion behaviors using transient tactile cues detected at the gripper's tip during insertion processes. These cues enable the robot to dynamically control and correct the 6-DoF pose of the grasped object. When combined with vision, TranTac achieves an average success rate of 79% on object grasping and insertion tasks, outperforming both vision-only policy and the one augmented with end-effector 6D force/torque sensing. Contact localization performance is also validated through tactile-only misaligned insertion tasks, achieving an average success rate of 88%. We assess the generalizability by training TranTac on a single prism-slot pair and testing it on unseen data, including a USB plug and a metal key, and find that the insertion tasks can still be completed with an average success rate of nearly 70%. The proposed framework may inspire new robotic tactile sensing systems for delicate manipulation tasks.