Once I put the racks back in there was no more water coming from the door. Like I said, it was not a small drip so it was an obvious change from putting the racks back in. The area under the dishwasher was always dry so it wasn't a leak from me replacing the motor.
We've done 2 loads so far and they have both worked. It's looking good so far, but I need to wait for a few more successful washes before I can definitively declare victory. Either way, I think it's safe enough to thank you for your help at this point!