Use carb cleaner.

While you have the cable disconnected from the carb you might as well hold it upright a drip a little light oil into the cable housing to lube it for smoother operation. Sewing machine oil, 3 in 1 or Dextron auto trans oil.
Dextron is my choice as it is more of a cleaner and will not make plastic or inner cable sleeves swell up and bind.

I have WD-40, Seafaom, motoroil, the gas in my tank, and various metal polishes (liquid and cream brasso, puma, neverdull wads) as well as typical household cleaners (windex, bleach... that sort of thing) I can buy other stuff, but I'm low on funds and don't want to buy things unless I really need them, so if there's something that would help now that I really should be using in my general bike maintenance anyway, then I'll get it, but I don't want to buy something just to fix an acute problem if something I have on hand would work alright.

edit: for the record, this was not in response to any single suggestion, just saying to keep it in mind. You guys keep saying to use things that I don't have...
Also, the carb sat in chem dip for a few hours (or whatever the recommended duration was) back in late November/early December, so I kinda assume/hope that did something as far as carb cleaning goes...

I WD-40 the crap outta the plunger hole in the carb, then pushed the screw up and used a screw driver to turn the plunger around in there a bit, after a turn or two, it finally came loose.
This video is kinda crap, but you can at least kind of see the plunger and how un-smoothly the choke cable is operating right now.
http://youtu.be/6XMsJQDRCno

You may be able to get it cleaned up to work properly.Also you need to disassemble the cable at the upper end & get some lube to flow down the cable.If this doesn't free things up you may need to replace the cable.But The plunger it's self may be the whole cause of the malfunction.Once clean & lubed it may work fine. ;) :)

How/where do you disassemble at the top and what would you use to lubricate?

I reassembled everything. It appeared that someone had at one point switched the overflow and air vent hoses. One's narrower, the other's wide, the narrow one had been stretched on to cover the higher outlet (the air vent), and the big one had been on the lower one (the overflow drain)... I switched them because it seemed like the small one fit better on the overflow drain than the air vent, is that right?
The choke's still difficult to open, but it is openable, but this still doesn't make my bike go.
What do I need to troubleshoot if the battery's good and it's turning over, but not starting?

There are two screws that hold the housing together on the bars.When you loosen then you will be able to see the choke cable.There are specific lubes for cables or you can use any light oil.I like to use chain lube as it will flow down the cable & than "jell" to stay in place.Your hose placement sounds right.I can't check for you as I no longer have my G.Z. I would open the drain on the side of the float bowel (on bottom of carb) & be sure you have fuel & go from there.

;) :)

I think I found some of my missing oil. I was reading various trouble shooting guides and one said to check out the air intake system, so took out my air filter and felt around inside the airbox for any foreign items and found black gold (ok, it was brown & translucent, but whatever! The floor of the airbox was coated in oil, and the drain hose was probably just about full of it. Shoo, there was almost as much oil in my airbox as there was in the oil pan (but that's partially because there was so friggin little in the oil pan)... that's probably a slight exaggeration, but it was still a mildly alarming amount.

It is normal to have a little oil/water in the airbox,it is blowby from the engine.There is a plug on the bottom of the drain tube to drain it.Just be sure the plug is reinstalled properly when you are done cleaning out the airbox.

;):tup:

Yeah, I did drain it, hence why I knew the drain tube was filled with oil. Wouldn't really describe it as "a little oil", but maybe it had built up over a long time, since I'd never done anything with the airbox before and I have no idea what POs have done