Here are your parts
Replacement parts for General Electric MSX27XLB Refrigerator | AppliancePartsPros.com

I would remove the evaporator coil cover in the freezer and check the coils.

If they have ice on them you may want to manually defrost the unit.

Could be that with the door left open the coils iced up so much that it will take several defrost cycles to clear everything.

While you have the cover off you may as well also check the defrost components.

First remove the evaporator cover in the freezer so you can see the coils.
Do not let them de-ice.
If they are heavily iced/frosted over you have a defrost problem.

If yes.
Manually force a defrost cycle by turning the defrost timer cam (Item 257 in Section 4) till the fans and compressor turn off.
Now check the defrost heater to see if it is on.
Be careful you do not want to burn your fingers.
If the heater is on then the heater and defrost thermostat are OK

If not on.
Unplug the unit.
Remove the wire for one side of the heating element from the wiring and measure it for continuity, usually around 20 ohms or so.

If the heater is OK
Remove one wire to the defrost thermostat and measure it, should be 0 ohms when frozen. Note that it opens just above freezing so must be frozen to check it. Also inspect it, if it is bulged at all replace it even if it measures OK.

If both the above are OK then odds are the defrost timer contacts are toast.
Best way to test this is a live test to see if you have 120 volts across the heater/defrost thermostat combo.

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.

Ok, I removed all of the contents of the freezer, including the shelves, and removed the plastic cover from the back wall, which revealed a bunch of coils. They have a thin layer of frost on them, but nothing excessive. I could easily scrape it off with my fingernail. And I can see the silver color of the metal coils, so it's not like their covered in frosty white ice.

So, should I still check the defrost components?

[COLOR="Blue"]So I guess this means the defrost components are working too.[/COLOR]
Yes it sounds like the defrost is OK.
Especially since the coils only had a light coating of frost. For defrost to cause a problem the coils must be frosted/iced over to the point where the evaporator fan cannot pull air through them.

Does the evaporator fan in the freezer run?
It is required for the unit to cool efficiently.

Was the frost coating on the coils fairly even?
If there was only frost in one area usually where the freon enters the coils or if it was very blotchy, it is an indication of low freon or a sealed system problem.

Is the condenser/compressor fan running?
This also can produce poor cooling.
Also are the condenser coils clean?

[COLOR="Blue"]How do I find/check the condenser fan?[/COLOR]
Item 650 in Section 5 of the parts.

[COLOR="Blue"] Is it accessible from underneath...[/COLOR]
You can get to it by removing the rear cover (Item 616 in Section 5)

Yes, you have to pull the fridge away from the wall.