I never played it myself, but I've heard of a adventure/puzzle Wii game called Zack and Wiki: Quest for Barbaros' Treasure. The name might sound stupid, but from what I've heard, it was a fantastic puzzle game. Also heard that it was really difficult towards the end, though...
I'm not recommending it since I haven't played it myself, but hopefully someone else on Wiichat could detail the game more for ya.
Easily some of the most famous adventure/platforming game series, you can never go wrong with the fantastic games Super Mario Galaxy 1 & 2. They aren't puzzle games like Zelda, but define the adventure game genre. Really fun, and not really difficult.
I would also highly recommend The Legend of Zelda: Wind Waker. It was a game for the Nintendo Gamecube, Nintendo's console prior to the Wii. Since it's quite a few years old, it's a very cheap but very enjoyable game. The reason I'm mentioning it is since you've played the Legend of Zelda game(s) on the DS, which are actually sequels to Zelda: Wind Waker.
While the game Red Steel 2 is an action game and doesn't have many adventure or puzzle elements, I would personally recommend it simply because it's a fantastic game, and may change your mind about games more combat oriented. Red Steel 2 is all about fast-paced fighting action via swordplay and shooting with your gun of choice; since you're not a fan of games that are strictly shooters, you'll probably like the fact you can forsake your gun through the entire game. I've gone through the entire game using only my sword, and to be honest, I thought it was more fun that way too. The game doesn't have any sort of incredible difficulty either. The game gives you as many tutorials as you need, and also gives you difficulties to choose from too, so if you're worried about having a tough time there's always easy mode. =)
Of course, if you are really against purchasing a game that's all about fighting and action, then I suppose you might want to ignore Red Steel 2. It really is that fun of a game though, hence why I bothered mentioned a game not pertaining to your video game tastes.