OK, I tried something else, and I THINK I've figured out what the problem is.
It seems that the way my headphone's are oriented in the hole is what's causing. I've got it rotated now such that the sound DOES output to the headphones and not the speakers, but if I twist it around at all (even putting on the headphones would twist it enough), it goes back to outputting audio to both the computer speakers and the headphones.
Is there some sort of composite I can put in the headphone jack to create a stronger connection or something? It seems like it's just not touching all the way or something, but I can't figure out how to fix it.
So, yeah, looks like a mechanical problem and not a software/hardware problem, which is good, I suppose.
Thanks for your help, and any suggestions on this most recent development would be greatly appreciated!