LG GR-602TVF Top Freezer

Try removing the fan/compressor cover panels and clear the path for airflow across the radiator. The fan and compressor should both run. Inside the fridge there is also a fan that should also run when the compressor runs. Airflow is a must across refrigeration radiators.
There is a defost timer which may need to be checked also. It cuts our the compressor/cooling cycle and turn on some heaters inside the fridge to defrost any ice build up. Common for these circuits to develop problems, the timer malfunctions.

low on refrigerant or compressor out

Clean back and bottom front of fridge vacuum all dust hair ect .see if fan is spinning freely

Your fridge operates like that: a compressor liquefies freon vapours; this creates heat so the hot liquid goes thru a cooling radiator with a fan on it (these 2 elements are at the rear bottom of the fridge) the freon fluid goes to an evaporator, turns to vapour and absorbs heat in the process, cooling the evaporator; a fan blows air over the evaporator and spreads the cooled air all over the fridge; the vapours return to the compressor.
In the rear of the fridge you should make sure that the radiator is clean and the fan working; inside the fridge you should make sure that the evaporator is not covered in ice and/or dirt and its fan is working. If necessary clean both.
If they are clean , the fans work but the evaporator isn't cold then either the freon circuit has sprung a leak and all the fluid has been lost or the compressor isn't working - to find out which just put your hands on the compressor : if it's hot and vibrates the fluid has been lost, if it doesn't vibrates then the compressor isn't working.
The compressor isn't working because either the thermostat or itself have failed. To find out which the compressor will have to be powered directly from the mains: if it starts then the thermostat must be replaced, if it doesn't the compressor will.
To fix the freon circuit or to replace the compressor a certified professional is mandatory, you must call one.

Wait a while untill the compressor cools down before plugging it back in. I don't think there is any problem with it. You obviously did not touch anything except clean the area.

Don't know the model/brand off the top of my head...

But: If the compressor is warm/hot, but nothing is happening, I ask this:
Even if the fridge is not cooling, is the compressor running, or dead silent?
If silent, but warm, do you hear a "Click" every so often ? Maybe 3 to 5 minutes apart, followed shortly by a 2nd click ?
If so, most likely the compressor has seized, and this is the over-current trip acting (thermal, so it needs time to cool).
Big problem if it is the compressor (expensive).

The only other thing that might cause a similar occurrence, would be if the compressor has a Start capacitor. If this has died, the compressor will not turn - just sit there getting warmer, as the over-current trip acts.

If it was a dead thermostat, the compressor wouldn't get power, and would be cold.

Most side by side units have 2 compressor units. If you listen for it, you should be able to hear 2 compressors with slightly different pitches, operating at different times, or sometimes at the same time. If you never hear the fridge side compressor turn on, a sensor or the compressor is probably bad, if you never hear the fridge side turn off, the temp sensor is bad. If it has ever, or is leaking in the back, then there is a puncture in the cooling system, and the fridge will not be able to be cooled. If you can pull the fridge out and examine the tubes or unplug it for about 10 minutes and plug it back in to observe compressor activity and any small leaks, it would give a more definite answer to the problem. A mechanic will most likely be necessary.

Check the back of the fridge and see if the coil is clean. If it is dirty then give it a wipe down. If the coil is clean then turn off the for two minutes and turn it back on listen for the compressor to start. If the compressor starts it is going to be a gas issue or electrical.

If the compressor does not start then you are up for a new compressor or something electrical. With out knowing anything more than what have you wrote, I would say to check these things and call a refrigeration tech.

Other than that I can offer support if you like

Sounds like you're low on gas...assuming that the condenser coils are clean.

Neither the freezer or refer sections are cooling? Can you hear if the compressor is running (black soccer ball sized unit underneath)? Could be "stuck" in the defrost cycle (happens a lot).