How do I do an “European out”?

You can receive a European out. There are many factors associated, many of which are restrictions placed by the German Government. You can easily google information related to these requirements. As for hoping for a US job on base, the US has initiated a drawdown, with that will be more base closures. DOD civilians will be fighting for those remaining positions. The competition will be fierce. For the Germans you must prove you are not a financial liability, they you will be able to pay for or have medical coverage. To stay beyond a certain period you must show language proficiency etc. having a girlfriend, or even getting married does not resolve these requirements. As said by others seek guidance of this living there, and check with the German authorities.

Check with the US Embassy and JAG

You will need to do some research on your end. Speak to S-1 and coordinate the military side. Make sure there aren't any loose ends on the military side. Speak to your nearest State Dept location whether that is a consulate or the Embassy. They should be able to point you in the correct direction for checking all of the German boxes for staying. If you are wanting to get a gov job start working on that now. It's a slow process. The good news is that even though some gov positions are reducing OCONUS positions are constantly in rotation because GS employees can only do OCONUS for 5 years before being rotated back to CONUS.