Depends on the state of his logistics and supply. Besieging armies can run short of food and ammo too.
You are right there needs to be a better explanation for the withdrawal.Banks also had cleared eastern Louisiana of confederate forces before launching his siege so their was no reason to stop from worries like that.
Yes they can, but that's not what was happening. Banks could easily supply himself from the river. In this TL the only reason given is a command crisis. If Banks just pulled out like that he'd be relieved of command, and face charges, which would end his military career.