OK, the electrical symptom stuff you are seeing proves your fridge fan is wired just like mine. (Sorta stupid-like by the factory, to be blunt).
The fan and it's thermostat are wired directly to the 12 volt
fridge power wire coming direct from your fuse panel, not the fridge control board or anything else itself.
As long as the 12 volt
power circuits in the trailer have power, the fan thermostat is the ONLY thing that turns that fan on or off. The little thermostat looks like a round black 'thingie' with a couple wires that connect to terminals on top of it. One wire comes from 12 volt
power, the other goes to the positive lead on the fan.
That explains the effect that you see, putting the use/store to "store" kills the fan, and that when the outside air temperature drops in the afternoon, the thermostat eventually automagically turns off the fan...the refrigerator on/off switch does NOT control the vent fan directly.
For storage, since I'm a bit lazy, I just go out, open the fridge upper door, and unplug the dumb fan, since the ambient air temperature around here in summer is often more than hot enough to trigger the fridge fan without the fridge being on. Hopefully I'll remember to plug it back in if I need to run the fridge.
Since I use one of the fantastic fans to ventilate the AS while it is in storage, and the solar panels efficiently charge the batteries, I store the AS in the "use" position. This way the vent fans can run, and I have lights if I'm working on the trailer...
A better way to fix it is to add a switch with a pilot light to the fan circuit somewhere convenient inside the coach, and just turn it off when you turn the fridge off from it's front panel switches. I could get real fancy-schmancy and hack into the fridge circuit board and figure out where thge fridge power switch turns things on and power the fan from there, but why risk it? Extra switches are cheap. (I know I'm risking losing my "Overkill Engineering Card" here, but sometimes just adding a manual switch with an indicator LED as a reminder is just easier.)
My "Overkill Plan" involves eventually replacing both my fridge outside cheap, ugly plastic doors with slick, shiny stainless steel louvered ones ($$$, ouchie), then adding at least two real quiet computer fans with a thermal speed control to the back of the top door, plus a handy manual cutoff switch inside.
That way, when it gets real warm in the fridge compartment during a hot day, the fans run full speed (unless I get a fire in there, and I have a safety thermostat in the planned circuit to cut it off at 140 degrees F for safety) and slow down the fans if it is not that warm in there at night so we can sleep.