Sounds like the kill wire that connects to the magneto isn't connected or isn't working. Could be a faulty coil in the aspect that the ground tab doesn't work but have never seen one. More likely not connected or something wrong with grounding circuit that is supposed to kill the spark when you get off the seat The key switch and the safety system both kill spark. Without that working it will not shutoff, and that 3-5 second delay when shutting off is the fuel solenoid shutting off fuel to the engine but not killing the spark.
Try removing the blower housing and with the engine running take a screwdriver and connect it between the kill tab on the coil and the engine block. If it shuts off then the issue is with the mower wiring. If it continued to run then the new module is bad.