1.  Yes, as far as I know a _care of_ address is sufficient.
    Note, the [address needs to be formatted like this](https://www.UPU.int/UPU/media/upu/PostalEntitiesFiles/addressingUnit/deuEn.pdf):
    ```
    Fr. Bằng Rikimaru
    c/o Hr. Brother XXX
    Hauptstraße 99 // App. 32
    Someville 12345
    ```
    _Appartement 32_ is the _Adresszusatz_.
    The apartment number is separated by a double-slash and part of the street name and house number line.
2.  The _nicht nachsenden_ note instructs the mailhub computer systems to
    
    * print the encoded ([_Zielcode_](https://de.WikiPedia.org/wiki/Zielcode#Deutschland)) inside address as is (thus perform no translation)
    * and _not_ print (and attach) a corrected human‑readable removable self‑adhesive label to the envelope.
    
    Therefore your mail should be (attempted to be) delivered to your “old” address.
3.  Undelivered mail is kept for seven working days at the local post office (or one of the many “service points” e. g. at the gas station).
    If it is not picked up, it is returned to the sender (if a domestic address is specified).
    There is no attempt made at delivering mail to your address in State A, that would be uneconomical.
4.  You book the forwarding service for a limited amount of time.
    Should you need to cancel or amend the contract before it expires, you are [supposed to contact customer support](https://www.DeutschePost.de/de/n/nachsendeservice/haeufige-fragen.html#:~:text=Wie%20kann%20ich%20einen%20bestehenden%20NACHSENDESERVICE%20%C3%A4ndern,%20stornieren%20oder%20k%C3%BCndigen?) specifying the order number.
    There are no refunds if the forwarding period already started.
5.  If it is feasible simply inform any potential senders of your whereabouts.
    Some senders will _always_ mail important letters with a _do not forward_ note, e. g. the broadcasting fees collectors _GEZ_, so you need to check (or let someone check) your mailbox anyway.
    For that purpose, there is also a [mail announcement service](https://www.DeutschePost.de/de/b/briefankuendigung.html) so you can learn about imminent deliveries during the night mail is processed for you (and esp. _who_ is sending mail if a return address is printed on the front).