The humming noise may be the evaporator fan either operating or trying to operate. It should be operating as long as the compressor is running and the doors are shut. It stops when a door is opened and starts again when it is shut.
Don't confuse this fan with the condenser fan which will run whether the doors are open or shut.
The condenser fan is outside the compartments near the compressor and condenser coils. The evap fan is inside the freezer compartment behind a panel
Can you hear or "feel" if the compressor motor is running?
Price may depend on whether the control board is the reason that the compressor is not working or whether it is just the compressor or both.
How old is the refrigerator?
Usually there is a 5 year warranty on the "sealed system" which includes the compressor (but not the labour), while the rest of the fridge is only 12 months which would include the control board.
Check the warranty on the fridge to see what it is for your model. It is usually found in the user manual
I find it surprising that both the control board and the compressor failed at the same time. Maybe if you had power supply problems e.g. surges, outages etc but otherwise very coincidental.
Either one failing only is more likely although the compressor should last for at least the warranty period. With electronics even though they are very reliable and robust these days occasionally they fail early
With regard to the compressor, according to this link (enter “compressor’ in the part name search box) the compressor for your fridge (part #300A in case parts diagram - part #TCA35911902) is not available any more and is no longer manufactured.
When searching online using the part number it turns out that the compressor is an LG compressor which does not bode well that it is no longer manufactured, as there was (is?) a class action lawsuit regarding LG compressors.
There’s more information online regarding the LG compressor problem
If you do have a LG compressor (verify and if so, also get the model number and serial number from the label on the compressor) maybe you could contact LG (see link at bottom of article in the previous link) and ask them if you feel that you might be caught up in this problem. That’s why knowing the model and serial numbers of the compressor might be handy. Just a thought.
It could be that a compressor fault damaged the control board.
If you want to test it out for yourself, here's a link to the service manual for the refrigerator, which should show how to get into the diagnostics to check individual components in the fridge, if the control board is still functioning that far that is.
The downside is that you have to purchase it.
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