I just need to get the final key piece to get near the ending. I got hit with the "turn in shield bug" so I've had a full list of locations and quests for quite a while so I know I'm almost done. I want to jump into the DLC before I beat the game because I'm assuming that once I beat the game I'll lose out on a playthrough of those DLC pieces since I can't repeat missions whenever I want.
My question is should I play them before I beat the game? I jumped into the zombie island one for just a second and I saw the cut scene talk about the hunt for the vault being over. Is the DLC meant to be played after beating the game?
I've heard some of the DLC gives special perks that could be useful during the normal game, but I've heard that playing the DCL will cause you to be overlevelled for NG+ (currently 33), which I don't really want to do. Also, is there an order I should play them in?