Disclaimer: Have not looked at one of these in person so all opinions are based on a semi-educated guess.
If it's a mechanical thermostat then the adjusting knob is most likely connected to a rheostat inside the white box. If you look inside the box it would most likely look like a small, round, metal box. Usually 1" more or less diameter.
I can think of two ways the knob could be frozen up and not turn.
1. Something around the shaft of the knob, most likely corrosion
, where the knob passes through the outside to the inside of the thermostat box or where that shaft enters the body of the rheostat itself inside the box.
2. Inside the rheostat the internal parts
could be corroded or there could be a broken or loose piece blocking the rotation.
IF the problem is due to corrosion
where the shaft of the knob passes through the thermostat housing then squirting some kind of goop might work
. Any of the other causes, spraying probably would do nothing.
I would try to remove the know. Should be fairly simple. Usually held in place by a small set screw or sometimes just a press on fit.
However, bottom line, a new thermostat is only $38.00 retail. I wouldn't spend a whole lot of time messing with this unless I was stuck in Tierra del Fuego and couldn't get a new one.