I went and rescheduled it the next day. They told me the farthest they could go is one more day... except after that... you pay any additional fees because that would be a voluntary change
This is the classic "hang up and call again".
Yes, DL can say you only have a one day window. But they would rather rebook you at 2 days than give the money back. Some agents will grasp this, some will not.
All that notification stuff does not matter at all. DL is at fault for not flying you from the original airport at about the original time. That is is. Their lack of notification is "bad" and you an be pissed but it matters for nothing. Their notification of the change does not make them innocent for trying to fly you out of the wrong airport.
Just stick to your guns with the agent.