I haven't seen the whole series yet, does she normally carry a large slug in there?

yeah, she's kinda infested with those littel dudes... too bad you only find out in the game, because the anime only covers 1/3 (Saber route) of the whole story, leaving Rin and Sakura's rout kinda out.

Does that have anything to do with Sakura being 'taken over by the darkness' late in the anime series?

Only indirectly.

In that, it's the cause of her being taken over by the darkness in the *game*, and the Sakura/Rin fight in the anime is a (much inferior) rip-off of the end of the Heaven's Feel route of the game.