Hazelbaker probably belongs in AAA. He's a fringe major leaguer.
I don't see how they can trade Adams without getting a corner position player back. Holliday and Moss are FA after the season (assuming Holliday's option isn't picked up, which would cost $16 M more than the buyout.) They have no other decent first basemen, and they will have an outfield need. When you look at the budget...it's difficult to imagine them wanting to have to sign two new OF and a new 1B. That's really expensive. I suppose you could move a Wong or a Carp, but that diminishes their value since they are decent infielders.
I can see the logic behind trading Wong for an outfielder. He has a reasonably favorable contract. But, on the flip side, the Cards don't have a need to dump him for nothing. They need to find something like the Craig/Lackey deal where they are getting something of worth in return.