Since this is a "new to you" trailer, the previous owner may have modified things. There is a very simple mod that can be made to the use/store that cuts the battery charger out when in the "use" mode.
On a typical trailer, the use / store works as you have described in terms of the lights and fridge. Shutting down the major electrical demand in the trailer is what it's there for.
Some trailers are setup so that the converter / charger works in both modes (use and store). Others are set up only to charge in one or the other setting. There is a wire you move between a couple of lugs on the relay to make this change.
First thing to check is the breaker that runs the converter charger. It may be tripped. Cycle it full off and back to on again to be sure it is correct.
Next thing to do is grab your multimeter ( everybody has one ... right
). The one that costs $20 on Amazon or close to that at Lowes / Home Depot will do just fine.
First up check the 120V into the converter / charger. The breaker may be bad or a wire may be loose.
Next check for voltage on the output of the converter / charger. If it's up above 13V, the charger is probably working ok. If it's down around 12V
more investigation is needed. If it's zero volts the charger has an issue and there is something else wrong as well.
If the converter is up at 13V, check on the battery posts. If they are also around 13V, the battery should be charging.
If the converter is at 12V
, cycle the breaker to turn it on and off. If the voltage stays the same both on and off, it's a good bet the converter is bad.
If you have zero volts, there is an open wire of fuse between your (blown) converter and the battery.
If your converter is working and the batteries die quickly, check the water level in the batteries. (only works for flooded cells not AGM's). You may find a dry cell.
One further complication in a modified trailer: There may be a full battery disconnect switch somewhere. This is a switch in series with the battery. It fully cuts off everything in the trailer. If it's in the "off" position, that's only for long term storage. It should be "on" any time you are using the trailer and whenever you are towing it.
So, lots of twists and turns. Indeed there are a few unusual ones not mentioned above. Hopefully this will get you moving.