I recently purchased all the DLC for Grim Dawn here. Do I need to start a new game or can I just start the game where I last saved without having to deal with a broken game?

Well The Crucible is just an endless wave mode that doesn't affect the main story. The other is just a collection of swag. So that leaves Ashes of Malmouth, which I have no idea since I don't own it. If it's anything like most extra story DLC, it's probably a late game chapter you can access on the map somewhere. But there's also two new classes, so you might want to start a new game anyway if you want to try them out.

Just continue your last save. The new content tacks on to the old without overwriting anything. Go to the Fort and see if anyone wants to talk to you.

Also, when possible, put game-specific questions in the game subforums, please.

Yes, it's an extra chapter tagged on the end of the game. It's not bad, didn't add an awful lot to my thinking. If you already have a character with your professions selected you would miss the extra classes, so it would just be the extra chapter. They also added loot, so you may find it difficult to get certain item sets only doing that chapter.