In some cases it's done with a secret combination of button presses and/or button holds on the front control panel. On my Samsung fridge it's like that. Inside the ice maker is a reset button, but no other internal buttons or switches to turn off the ice maker. I have to do certain ctrl-alt-del style button presses on the front of the fridge to get it to shut off the ice maker. I'm sure they're in the manual somewhere but I'd just search the internet for instructions.

I wonder if some of that unwanted ice is mostly just the leftover water in the ice maker tubes (and the pump and the filter canister) slowly leaking out over time after the fridge got moved.

If it's humidity, there may be instructions on how to correctly plug up the water-input hole so that humidity doesn't get in.