Around halfway through Mission 5 the game has started cutting to a black screen after I get onto an elevator and doesn't load the next floor. After the first time this happened, the other elevators I can access all produce the same bug. Reinstalling has had no effect.

Everything else seems ok, and I started a new game to see if it would load fine, and it did. Anyone have any suggestions on how to troubleshoot this? I'd rather not have to enable cheats in order to level warp because it overloads you with credits.

It could be savegame corruption, i've heard it may happen with Crusader, though i've never witnessed it myself.
Perhaps, as a safety measure, You could use a few slots for Your quicksaves?

Before getting into the elevator have you noticed that the screen did not 'move' automatically? I had such a thing before this 'bug' happened. Just load an earlier save game.