Here's a how I did it vs. a how to do it. If you choose to do similar it is at your own risk.
On my rig I added two relays to controll that the 12VDC plug only gets power if the 120VAC is without power or disconnected. This isolates the 12VDC circuit if 120VAC has power as to prevent the particular 12VDC backfeed to other downstream 12VDC cicuits because of the defect in the Dometic cooler controller board as called out in the recall. IMHO they should recall/replace these boards, but it appears they got away with a "do not plug in both 120VAC and 12VDC at same time" sticker.
I did the work such that I could go back to how the factory wired the 12VDC outlet by simply moving the 2 factory connections in the 12VDC outlet box back to the socket, which disconnects my mod. A 120VAC sense wire plugged into a 120VAC outlet splitter in the bay powers a 120VAC relay. If the relay is powered it disconnects 12VDC trigger voltage to a second 12VDC high amp relay. That second relay only powers the 12VDC socket the cooler plugs into when there is no 120VAC present on the first relay. If I had my druthers I would have made the mod inside the cooler itself vs the RV storage bay, but this was easier...
On my unit, if the cooler is already powered on at the time, it seems to stay on (after a quick delay) as power goes back and forth between 120VAC and 12VDC. Yay!
120VAC outlet splitter:
Amazon.com
120VAC relay that can switch 12VDC to power the 2nd relay:
Amazon.com
12VDC high amp relay with built-in fuse:
Amazon.com
Pics: installed mod, mod jumpers to leverage factory 12VDC outlet without cutting or splicing into factory wires and easily return back to factory config, closeup of relays mounted next to 12VDC outlet box.
Best,
-Mark