The solution is either to make occasional position updates or to use stepper motors with sufficient accuracy to carry the robot through the maze without collisions. Dead reckoning with DC motors instead of steppers is much more difficult, even with wheel encoders. Another way to navigate through the robot house is to use sensor feedback to sense and avoid the walls. You can combine wall sensors with rough dead reckoning to indicate the end of a hall or the presence of a doorway into a room. Recommendations Many options exist here.
Steppers also are low torque devices.
Design the robot size and shape to give margin for error. Instead of a 12-inch square, aim for something 6 to 10 inches in diameter and try to keep it light. A light robot will be better able to accelerate and turn. While placing the drive wheels on the centerline isn't necessary, it does allow tighter turns. If the robot dead reckons, precise turns are essential.