True, but he knew the Raiders were going to cut him so he would be a free agent. So anything he did would be to the Raiders advantage and not his. Here he knew that the Saints were not going to cut him due to the cap hit and Moore wanting him to stay. But, there could have been some advantages to him if he could get another team to give him a longer term contract with more or the same money. Geno Smith is reportedly about to get $40 or $45 million per year from the Raiders. That probably boosted what he would take to stay with the Saints or what he would have needed to see in a contract with another team before he would do things to make a cut or trade feasible.
So, if he really wanted out, he would have had to demand a trade and likely restructure to make it work under the cap. But, he did neither of those. I suspect the truth of it was that he would have considered a trade if the Saints proposed one and he wanted to have permission to gage what kind of contract he would get, but was also unwilling to change his contract to make a trade or cut feasible for the Saints. So, I'm sure he was interested in other options, but he wasn't so interested that he was willing to do things to make a cut or a trade happen.
I also suspect that in the end Carr realized that no team was going to be willing to pay him what the Saints were paying him and since the Saints were not going to cut him, he would hold his ground. It's all about the money for Carr, as it should be, so it's not so much that he wants to be here, it's that he likes the contract he has here and isn't going to change it.