It's Doom95 i'm using. It's the Collectors Edition with all three games on it and the modern menus.
I tried all these things and nothing works. I've tried it with another joypad, i've reinstalled everything possible, i've pressed every single button during play, explored everywhre in the menus, i've even e-mailed Activition who didn't help and i now know for sure that there is NOT an always-run option. However, i tried playing an older version of Doom and it worked fine, so there is definitley nothing wrong with the joypad.
I'm using Win98 by the way and sorry for the delay, cheers.