Had to fight with my Dometic a couple of years ago, fried control module was the culprit. Mine, (93 model) does not have a slider.
The fridge temp is dictated by the freezer temp.
It simply fires up more often to run colder. Flame is fixed by an orifice. Never use the AC mode.
I found for the first fire-up of the season, level is critical. After that, (every couple of weeks) it is more forgiving.
I went to the Dometic site, got the manual, and the name of a close dealer. They had the parts I needed on the shelf.
Parts were not cheap, $400, but better than the price of a new fridge, and tearing half of the trailer apart.
I will have to try the ice-cube trick.