Tim-
Now that bstireman mentions it, because it's been nearly 2 years since we were on the road, my memory is jogged that I had to learn to turn off the icemaker lever on our Dometic 2-door also, before hitting the highway. In fact, as I recall, it was best to do it the night before so there was no chance of there being a fresh cycle of unfrozen water in the tray when we drove off.
The refer's timer automatically defrosted every night, but there was still ice and frost on the mechanisms, so I used to key in one or two extra defrost cycles manually during the day, but to little avail in removing the aberrant ice buildup. Then I surmised it might be from slopping water that hadn't had a chance to freeze, and lifting the actuator wire at least several hours before traveling did help.
From your description, though, Tim, your box is overflowing when it fills. I think I read in our Dometic manuals somewhere that there is an adjustment for setting the fill point. There is likely a float or weight switch somewhere that shuts off flow at the set point of fill. If your Norcold is adjustable, that may be your simple fix. Hopefully the switch isn't faulty. If you don't have a reference to it in your Norcold documentation, you might check online at Norcold's website.
-Joel