Here are your parts, I think.
I cannot find info on the given model number but the below is for a BAWW not a BWW. I am assuming you just missed the A.
Parts for GE TFX20JRBAWW Refrigerator - AppliancePartsPros.com

[COLOR="Blue"]It is not airflow (I don't think)--we didn't have too much in there.[/COLOR]
A defrost problem is actually an airflow problem. The evaporator fan cannot pull air through the coils so the air does not cool correctly. Also it screws up the amount of air that is pushed into the fresh food section.

[COLOR="Blue"]--in fact, I think they stay on all day it seems.[/COLOR]
The compressor and fans running all the time is one symptom of a defrost problem so I think you are on the right track.

[COLOR="Blue"]Also, the ice maker doesn't make ice, but I'm guessing that's because it's not cold enough in there.[/COLOR]
Also correct.

[COLOR="Blue"]So from my research it seems most people suggest it is the defrost timer, the temperature control, or the thermostat.[/COLOR]
Possible causes of a defrost problem is the defrost timer, the defrost thermostat or the defrost heater.

[COLOR="Blue"]My question is (are, actually): if I stick the screwdriver in the defrost timer, turn it, should the fridge then turn off?[/COLOR]
Correct.
The compressor and fans should shut off, the defrsot heater should come on.

[COLOR="Blue"]If so, then I suppose I just change the timer (is that correct?).[/COLOR]
Sort of correct. If the defrost heater comes on then you problem is the timer.

[COLOR="Blue"]What else would you all recommend? [/COLOR]
Below is my standard blurb ob troubleshooting defrost problems, I just added info on the parts.


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

If yes.
Manually force a defrost cycle by turning the defrost timer cam (Item 257 in Section 07) till the fans and compressor turn off.
There is usually a hole in the cover to let you do this without taking anything apart.
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 timer needs replacing, re: it is probably stalling during it's rotation so is never getting into a defrost cycle.

If not on.
Unplug the unit.
Remove the wire for one side of the heating element (Item 230 in Section 04) 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 (Item 240 in Section 04) 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.

[COLOR="Blue"]the fridge power dimmed a bit, but that's all--everything continued as before: the compressor fan was on, the freezer fan kept spinning.[/COLOR]
There is a problem here.
If the timer cam be turned through 360 degrees and the unit does not go into a defrost cycle then the timer needs replacing.
For it to be in defrost the fans and compressor must be off.

[COLOR="Blue"]If the part is busted, shouldn't it spike?[/COLOR]
No.
If the heater is open/busted the meter needle will not move re: infinite resistance.
Your testing method seems OK.
You could try a higher meter setting but I think you may have two problems.
The defrosts timer and the defrost heater are toast.

I wou;ld aso check the defrost thermostat just in case it also has a problem.

I have not used an analogue meter in over twenty years so do not take what I say as gospel.

I agree that the timer is bad.

I also think that the defrost heater is bad.
No needle movement means no continuity so the heater is open (blown).

I am not sure about the defrost thermostat.
I do not have and cannot get a wiring diagram for this unit. GE likes to keep everything a big secret.
If you trace the two wires going to the defrost thermostat back to the plug and they measure 0 ohms on the meter when the defrost thermostat is frozen then it is OK. 0 ohms is the same as when you short the two meter leads together.

Note that the defrost thermostat opens just above freezing so it must be frozen to see the 0 ohms. If the evaporator coils in the unit have no frost/ice on them then hold the face of the defrost thermostat onto an ice cube.

Hello Denman, unfortunately, things aren't fixed. I must have done something wrong, or there's something else broken.
So I replaced the defrost timer first, nothing changed, though now I can use the screwdriver to turn off the refrigerator--so I presume my old one was bad.

I installed a new defrost heater, and it came with a new thermostat, which I installed. Unfortunately, when I use the manual defrost with the screwdriver, the heater does not turn on. Frost is starting to form in the freezer (I had thawed it).

I think everything is installed correctly. Presuming my replacement parts were fine, any thoughts what could be the problem? Thank you.