Might I recommend Condemned... that game is simply amazing... easily my favorite Xbox 360 title.
It's an adventure/horror title but it's got an awesome storyline. The graphics are fantastic as well. It's kind of a short game, but it should only be about $20-$30 these days because it was a launch title (though a total sleeper hit).
Oblivion is another good choice, massive and expansive story, the game can easily take 100+ hours to fully complete. Excellent graphics here too.
Other good choices are Dead Rising, really massive game, very interesting story, and once again fantastic graphics. This one will take you quite a long time to fully complete.
Gun is a good game, the story and gameplay are excellent but the graphics are so-so compared to your other choices. The game length is pretty good though, not too short, not too long. maybe about 15-20 hours unless you're rushing through it.
Kameo is a really fun game, excellent graphics and gameplay but it's a little weak in the story department, also a fairly short game
I'd have to disagree with who ever suggested Gears of War, while it's a fantastic game it doesn't sound like a very good match for what you're looking for, it's a shooter not an action/adventure, it's got next to nothing for a story and it's incredibly short unless you're playing multiplayer. It's a wickedly fun shooter, particularly for co-op and multi but it's definitely not an action/adventure with a deep story.... Just because a game is good doesn't mean it's good for everything and everyone.
Assuming you have $120 to spend I'd say:
$10 NHL2K6
$30 Condemned
$20 Kameo
$60 Oblivion or Dead Rising