I suspect it has to do with the internal wiring of the driver IC. First I would try to ground one of the (non-DP) segment resistor and see if it still comes on dim. E.g. disconnect the resistor from Out 1 and tie it directly to ground. I assume that segment would not come on dim under the condition you described, which would confirm the current is coming from the driver. I think it's probably obvious that it is, but it would be good to confirm, and rule out other strange possibilities, e.g. faulty components, wiring shorts, etc.
Assuming the current is coming from the driver, I would suspect the diodes on each output. These are great flyback diodes for driving relays or other inductive loads, but what could be happening here is that the ground pin gets a high voltage transient, which can pass directly through all the outputs. I assume you don't have long cables or other sources of inductance, so I would guess then the ground for the IC and for the MOSFET are not sufficiently low impedance. Perhaps try to connect the source of the MOSFET and the GND of the ICs directly to power source, if not already, rather than tying them together first.
That being said, it seems like a long shot to me but is based on the assumption that the driver inputs are indeed staying low. Perhaps there are brief pulses, noise, etc. at the input pins. If you aren't sure of that, then I would first try disconnecting the input from one of the segment drivers (tying the pin to ground), to be certain the input is actually low, and see if that segment comes on dim.