I had same problem with a freezer going down to -25°C and motor never turning off.
I phoned around and no one had a shop only option is a visit and the call out charge was £60 and freezer cost £100.
So next was hunt for a new thermostat and the one for the model number had a tapped fixing hole and mine did not so clearly wrong. So next step was to remove thermostat and inspect. It was reasonable easy to remove and it had a part number printed on it.
However I could not find a match for sale so selected one as close as I could find. When it arrived found shaft too short and it was even at maximum slightly over temperature.
The second problem was measuring temperature. Freezing point of brine is -18°C so finding too cold was easy but my attempts to find a thermometer that would work in a freezer failed.
However I wanted to brew beer so I bought a MH1210A temperature controller which went -40ºC~120ºC ; measurement error: ±0.5ºC and used this to measure temperature.
Clearly the same device could control the freezer and I did consider using one as the temperature controller.
but as you can see from the picture it does not come mounted and the first job is to house the unit.
As an external control unit one can lift the temperature of any fridge or freezer but there are other considerations. The old fridge/freezer I use in the garage has an electronic control and the unit uses a sensor rather than a simple tube and so to repair becomes more of a problem.
With my in use fridge/freezer it has solenoids and fridge and freezer are independently controlled. It also has an inverter controlled motor and is not suitable for DIY repair.
Clearly some skill is required and we have no idea of your skill. So although replacing something like this
is not too taxing they are not all so simple.