Yosef Karo

“ Rem"a: This is also true when Rosh Hashanah falls on Shabbat. The end of the blessing mentions only Shabbat. The same is true when Yom Kippur falls on Shabbat (Mahari"l and Minhagim) .
סעיף ג
בשבת בינתים ערבית יאמר בברכת מעין שבע המלך הקדוש. הגה: והוא הדין אם חל ראש השנה בשבת. וחותם בשל שבת לבד, וכן כשחל יום כפור בשבת (מהרי"ל ומנהגים) .
582:4 Some have the practice of praying on Rosh Hashanah and Yom Kippur bent over. These people need to straighten up at the end of each blessing. ”

“ Shulchan Aruch Orach Chaim 582 Prayers on the Ten Days of Repentance and Rosh Hashanah
סדר תפלת עשרת ימי תשובה וראש השנה
582:1 On the Ten Days of Repentance, we say "the holy King" and "the King of Justice." If one erred or has doubt [about having said the correct formulation] , with respect to "the Holy King" they return to the beginning [of the Amidah] , and with respect to "the King of Justice," if they realize their error before moving their feet [at the end of the Amidah] , they return to the blessing "Restore..." ”

“ This is the case even if they have not moved their feet [at the end of the Amidah] and have simply finished that blessing [that contains the addition] (Terumat Hadeshen) . The same rule applies to somebody who did not say "write..." ”