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I've used the differential_drive package as you suggest. It wants absolute counts reported by the robot controller, not relative. I assume you're planning to run your motors only forward. I've successfully used non-quadrature encoding while running the motors both directions, but it requires coasting to a stop, waiting, updating some state variable, then reversing direction. The total encoder count can be incremented or decremented, depending on the direction state variable.

However, I've never tried to do bidirectional movement without quadrature encoding while using ROS. If you use the ROS navigation stack, you necessarily relinquish some low-level control of the motors to gain the benefit of the global and local planners. Both the planners assume the motors can move either direction.