Cooling turning off (lights/display working) new fridge doing same!

We've had a Samsung American style fridge/freezer for 12 years (don't know the model number), in the last week it started to turn the cooling and freezing off, whilst the lights still worked and the display still showed the desired temp setting. After turning the socket switch off then on, the cooling would work again, but only temporarily, and then it would turn off again.

Just got a new fridge (Haier HRF-628IF6 American Style) the other day and it is doing the same thing. When it does work, the temp averages on about 5C to 7C, even when setting it to 1C (tested with several thermometers).

Was wondering if there is something tripping it in the power supply, rather than the same fault with two different fridges. Checked the sockets with a tester and it says it's fine. Tried plugging the fridge in another socket that is on a different circuit – still turning off!

I've got a wine cooler/fridge in another room on the same circuit, and that's been fine (colder than the new fridge when it works). Please help, I can't figure out what could be the problem. Thanks!