After experimenting some more, it seems to primarily be an issue with my joystick right now. As I said before, it's a few years old, and there was a period of a few months where I didn't even use it, so its spring is a little looser than it should be. If I wiggle it about a fair bit and hold it to the right, I can get it to stay mostly straight until I nudge it again, then I start veering left as always.
For a while I was considering investing in a whole HOTAS setup, but those seem pretty expensive, so even just a normal joystick would probably be an improvement.
That said, a deadzone setting would still most definitely be appreciated, since joysticks are gonna wear down sooner or later, and it'd be annoying having to buy a new one every time just so you can play Oolite without veering off to the sides. :p
Somebody posted elsewhere a £30 HOTAS system from Thrustmaster...