OK I tried replacing the existing driver and it didn't really make a difference, but then I found out in the particular game I was trying to play, that it disables joystick support by default because it crashes some versions of 16-bit windows. I turned that on in the game's INI file and now the joystick option is no longer greyed out and can be selected.
HOWEVER,
Even after properly calibrating the 2-axis joystick, it constantly has input to the left and ignores the Y axis. All I can do is make it stop turning left or allow it to, nothing else on the gamepad thumbstick seems to register. The buttons both work fine though.
UPDATE:
To fix the problem, I only had to click the SETUP button within the game, where it prompted me to center the joystick and press ENTER. I did so, and now it works perfectly! Not sure if changing the DOSBox settings for timed joystick to "true" helped or didn't matter, but it works well now.
Thank you, the guide was helpful, but my solution had mostly to do with the game itself not being set up properly, it turns out.