+1 for the train, but I would suggest book Rail & Fly if possible.

This has to be booked through your airline (American) and not all offer it, but then you can get a single or return ticket which is unlimited and for a cheap price, So if your flight is late no problem! But it is not clear for me if American offer this just for passengers originating in Germany or not, so you might want to ask them.

link in German
https://www.aa.com/intl/de/specials/...false&from=Nav