There is no direct analogy of the English phenomenon in German. If you just look for old-fashioned manners of addressing someone, using the second person plural pronoun Ihr for a respectful way of addressing someone (Was wünscht Ihr?), instead of the third person plural Sie which is used in modern German, and using the third person singular personal pronouns Er and Sie in order to address a person of lesser social status (Was will er?, Was will Sie?) will make your text sound more archaic.
If you look for more general ways to "archaize" the German translation, here are some inspirations:
In the novel Doktor Faustus by Thomas Mann, the devil sometimes speaks a language that is meant to resemble the language of Luther. Off the top of my head, I just remember constructions using elisions such as weißtu (instead of weißt du) and hastu (instead of hast du). You could look into the novel in order to study the way Thomas Mann did this in more detail.
I don't know if this is actually Lutheran German. It is not improbable that Mann just invented some formulations. If you don't trust Thomas Mann you could do what he probably did as well: you could study the Luther Bible, there are scans of the Old Testament and the New Testament.
Just to make an example I have looked into Luther's introduction into the Old Testament as linked above:
Das Alte Testament halten etliche geringe, als das dem Jüdischen volck alleine gegebene, und nu fort aus sey, und nur von vergangenen geschchten schreibe, meinen, sie haben gnug am Newen Testament, und geben für eitel geistliche sinn im alten Testament zu suchen, wie auch Origenes, Hieronymus und viel hoher leute mehr gehalten haben.
Aber Christus spricht Johannis am funfften: Forschet inn der Schrifft, denn die selbige gibt gezeugnis von mir. Und Paulus gebeut Thimotheo, Er solle anhalten mit lesen der Schrift, und rhuemet Roma I wie das Evangelion sey von Gott in der Schrifft verheissen, und in Corinth XV sagt er, Christus sey nach laut der Schrifft von Dauids gebluete komen, gestorben und vom tod aufferstanden.
Already in such a small snippet you can find some useful patterns:
- using the et ending in verb conjugation (forschet instead of forscht). To some extend, it still exists today. But most often it sounds archaic. Applied consistently, this could be a means to make a German text sound old.
- using the old Dativ form of (some) nouns with -e which is gone missing in modern German: auf dem Wege; in dem Hause; zu Bette gehen;
- old verb forms such as gebeuen, modern gebieten. (Similar would be dräuen instead of drohen.)
- using the y instead of i in sein: sey. You might have a lot of chance to use this, as sein is a frequent word.
- Using the Latin declination. In the snippet: Luther uses the Latin dative form of Timotheus, Timotheo. In modern German, Latin declination is only used for the nominative plural, with only few exceptions, most prominently the name of Jesus. For the name of Jesus, Latin declination is accepted for the Genitive Singular, but not for Dative or Accusative Singular (It is OK to say die Jünger Jesu, but not meinem Jesu or meinen Jesum, only meinem Jesus and meinen Jesus would be considered correct.) Similarly, Luther uses Latin declination for the dative of Johannis. Today only Johannes would be correct for the dative form, while Johannis is only used in Genitive, and hence in derived names (Johannisstraße, Johannisplatz, Johannisbeere).
- funfften - writing double ff after n, a combination that does not exist in modern German. Also, Schrifft has a double ff which does not exist anymore.
- the greek ending in Evangelion instead of the latinized ending as is common in modern German (Evangelium)
- using prefix für instead of vor (fürgeben instead of vorgeben).
- using the prefix Ge- in nouns where it is not common any more (Gezeugnis would be Zeugnis in current German)
Texts from the late medieval or early baroque era might be
another source of inspiration. Prominent authors are Andreas Gryphius and Paul Gerhardt. You will find texts written by them online.
Most prominent examples of "conserved" antiquated language can be found in chorals composed by Johann Bach, and in old Christmas carols. The language is conserved -- people recognize it as antiquated, but changing them would be considered to big of an intervention with the work of Bach (even though he didn't write the texts). I remember the christmas choral Heut schleußt er wieder auf die Thür (Instead of Heut schließt er wieder auf die Tür.).
Another source of old stages of German are dialects. Dialects are roughly snapshots of older stages of the German language. Hence dialects can be a source of inspiration. If you do not want to risk your language to sound like a particular dialect, I would look into dialects from German speaking communities outside of Europe. In Germany, they are less known than the european dialects of German, and hence the chance that they come across as "old" is higher in a German audience.
Finally, to mention another example off the top of my head: In texts of Arthur Schopenhauer, I found that verbs originating in romance languages which end on -ieren are written with he suffix -iren, emphasizing the Latin or Roman origin stronger. So, Schopenhauer would write affiziren instead of modern affizieren.
This answer is a community answer. If you have more ways to make German texts sound archaic, feel free to amend the answer.