I thought I'd come back to this topic to officially announce that I'm dumb!
I opened both joysticks, cut the cord off the one I assumed I killed and connected all the wires to the one I found in the trash. Plugged it into my little gameport to USB adapter and it behaved exactly the same as the other one! I had no proof that the one from the trash was good, but it seemed strange that it would have the exact same problem as the other one, which I was sure I caused by using a meter to check the connections.
So for the hell of it, I plugged it into my soundcard. Guess what; It worked perfectly.
Apparently they're not compatible with the gameport to USB adapter. So there was probably nothing wrong with the other stick, it just didn't like how it was plugged in!
Now I need to put a new cord on the one that originally had a cord. Unfortunately, I cut the wires off as close to the (soldered) connector on the board as I could, so I didn't leave myself anything to connect new wires to.
If I try to unsolder the pins, I'll probably make a mess of it. Maybe I can cut the plastic housing away from the pins and that will leave enough to solder to.
So yeah, I should have thought of trying it connected to my sound card before I jumped to the conclusion that I had killed it, and cut off the cable.