The game (single-player) has been running pretty well thus far (only the occasional crashes that fix themselves with a reboot), but now that I am entering the swamps, it gets excessively choppy, and then crashes entirely. I'm guessing it's a mix of the rain effects coupled with all the other swamp graphics... but it's crashing every time and I'm unable to finish the game.

I'm running the game on a 1.2 giga-hurts Athalon with 256 ram, and the video card that came with the game (I didn't install any voodoo card, or catholic card or devilworshipper card or anything), and I haven't installed the new patch (and would prefer not to, if given the choice).

try lowering your settings it may get better. What graphics card have you got?

I had a very similar problem - the game started to crash in a swamps without an obvious reason, but I was able to make the level through! Probably, the problem is - except of the graphic's card drivers compatibility, a bug in a level itself that is caused by the rain effect. In my case - the game was crashing mostly in a spot, near the beginning of the level, when Kyle has to jump on a high cliffs, and is - right on a top- attacked by two Jedi's in Shadow Armor. When I was trying to engage either one of them, the game was jumping to Windows, and I was unable to make a level through this point. If your game is crashing in a similar place here is what you do:
First of all - DO NOT engage any of the Jedis. When you approach the place use the Force speed and RUN forward like hell!!Afterwards do not try to go back, 'cause the game will also crash. Just continue through the level. Afterwards my JKII game crashed two or three times more in a swamps, but that was not a bigger problem.