The Kitzur Shulchan says to say Hashem and not Adoshem which has no meaning. When we referred to Hashem in Hungary, we said Hashem Yisborach as indicated in the Sefer Hachasidim based on the pasuk זכר צדיק לברכה. The Rosh for this reason instituted to say, ברוך הוא וברוך שמו.
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