Temperature control bellows

If you do use a modern thermostat, it's essential to include an external overload protector, since this is part of the original control. This is especially so with CA machines, since stalling on start up, if the NCG build up gets too much, is part of life.
The bellows should have pressure on them at room temperature and look like this if you remove them from the control www.flickr.com/photos/13469158@N05/7984118671/in/album-72157633054818403/
Beware that removing the bellows at room temperature is not recommended since it will put the control out of calibration, and there is some risk to damaging them.

Seeing as the compressor runs, this would indicate the bellows are probably OK, since it's their pressure that causes the contacts in the control to close. However, as you've had them out of the control at room temperature, they will have expanded to the point of permanently stretching the brass or copper, or whatever it is they're made from. So, in the least, recalibration is likely to be required. This is not difficult and is explained in the service manual.
One thing to check is the sensing tube is clamped to the evaporator in the correct location. If it is only sensing the cabinet air temperature rather than the actual evaporator, the fridge will run longer than it should (and thus run colder). The action of the bellows can be checked by freezing the sensing tube. The manual says to use dry ice, but a freezer is probably easier. The other thing to check is the thermostat mechanism actually works - do the contacts open and close when pressure is applied and released at the point where the bellows make contact?