Hi @pingwinator 

You can order a move from Telekom.
Ideally, your connection will move 1:1.
On the day of the move, your old connection goes offline and the new one starts.
But with a week it would be very short-term, because on average it takes 2 - 3 weeks.

Please refer: https://www.telekom.de/hilfe/umzug?samChecked=true 

Even a new contract takes 2-3 weeks.
But you then have to pay for the old connection until the end of the contract.

pingwinator

Can I order a new connection in advance without disabling the existing one for a short time?

Yes, you can.

The key word is "überlappender Umzug" (means overlapping translocation), the following wiki article in German

https://telekomhilft.telekom.de/t5/Festnetz-Internet/Uberlappender-Umzug-voruebergehend-zwei-Anschluesse-parallel/ta-p/5788194

Best you get that done with the help of a forum Teamie . As you have filled your forum profile - hopefully also indicating time intervals whe you can be called - be prepared to state the last 6 digits of your IBAN bank number when you recevie an incoming call from a Telekom Teamie .

The answer given by @→Mataimaki← focusses more on the normal translocation situation, but it gives an important hint that it may take two to three weeks after ordering until you will get a connection in your new place.

I want to say a huge thank you @Louisa G. !
A technician came last week and set up a new connection. Now everything works