We study the problem of using high-resolution tactile sensors to control the insertion of objects in a boxpacking scenario. In this paper, we propose an insertion strategy that leverages tactile sensing to: 1) safely probe the box with the grasped object while monitoring incipient slip to maintain a stable grasp on the object. 2) estimate and correct for residual position uncertainties to insert the object into a designated gap without disturbing the environment.Our proposed methodology is based on two neural networks that estimate the error direction and error magnitude, from a stream of tactile imprints, acquired by two GelSlim fingers, during the insertion process. The system is trained on four objects with basic geometric shapes, which we show generalizes to four other common objects. Based on the estimated positional errors, a heuristic controller iteratively adjusts the position of the object and eventually inserts it successfully without requiring prior knowledge of the geometry of the object.The key insight is that dense tactile feedback contains useful information with respect to the contact interaction between the grasped object and its environment. We achieve high success rate and show that unknown objects can be inserted with an average of 6 attempts of the probe-correct loop. The method's ability to generalize to novel objects makes it a good fit for box packing in warehouse automation.