1. The Order of Prayers for the Ten Days of Repentance and Rosh HaShana. And in it is 9 paragraphs. In the Ten Days of Repentance (Asereh Yemei Teshuva) it is said The Holy King (HaMelech HaKadosh in the conclusion of the 3rd Blessing of Amidah) and The King of Judgment (HaMelech HaMishpat in the conclusion of the 11th Blessing of Amidah). If one made a mistake or even if one is uncertain (if one said or not), (the rule is as follows:) If it is with "HaMelech HaKadosh" he should go back to the beginning (of Amidah, that is repeat). If it is HaMelech HaMishpat if prior to uprooting his feet (that is prior to finishing the Amidah by taking steps back), he should go back to the Birchas Hasheveinu (the 11th Blessing which concludes HaMelech HaMishpat) and say from there and on according to the order. If he did not remember until he uprooted his feet [see before the end of Siman 117] he should got back to the beginning (i.e. repeat). [Also see Siman 118.]
2. If one said The Holy God (Ha-El HaKadosh, the usual conclusion to the 3rd blessing) and within a moment (that is the Halachic time of "Tok Kedai Dibbur" - the time it takes to say a short phrase) he remember and says The Holy King (HaMelech HaKadosh), he does not need to go return (the beginning). The same rule applies with The Holy Judge (HaMelech HaMishpat).
3. In the Shabbos between (R'H and Yom Kippur), at Maariv say in the Birchas Mein Sheva (the added benediction after the Silent Amidah which represents the different blessings of the Amidah in short form) The Holy King (HaMelech HaKadosh). Note (Rema): So to if Rosh HaShana was on Shabbos but the conclusion is Shabbos alone. And the same is if Yom Kippur was on Shabbos [Maharil and Minhagim]
4. There are those who have the custom to pray on Rosh HaShanah and on Yom Kippur while bowing; it is necessary for them to straighten up at the end of the blessings....
5. If one did not say "Zachreinu" "Remember Us"(the added benediction in the 1st Blessing of the Amidah) or "Mi Kamocha" "Who is like You" (in the 2nd blessing), one does not return to it. Note (Rema): Even if one did not uproot their feet yet, rather that the Blessing was completed [Terumas HaDeshen]. So to if one did not say "v'Kesuv" "And write" (in the 18th blessing) and "uvSefer" "and in the Book" also the rule is so (that you do not go back even if just concluded that Blessing itself) [Tur]
6. And in the Prayers (of Rosh HaShana) say "vTiten Lanu Es Yom HaZichron Hazeh" "And He gave to us this Day of Remembrance", and do not mention Rosh Chodesh.
7. If (Rosh HaShana) is during the week, say (in the Amidah) "Yom Truah Mikra Kodesh" "A Day of Blasting, a Holy Convocation". If (R'H) feel on Shabbos, say "Zichron Teruah" "a Reminder of Blasting".
אינו אומר מועדים לשמחה חגים וזמנים לששון וכן אינו אומר והשיאנו ובתפלת מוסף אינו אומר ואין אנו יכולין לעלות ולראות לפניך אלא ואין אנו יכולים לעשות חובותינו לפניך (וחותמין ודברך אמת וכו') (טור):
אע"פ שכל ימות השנה מתפללין בלחש בראש השנה ויום הכפורים נוהגין לומר בקול רם ולהטעות לא חיישינן כיון שמצוין בידם מחזורים: הגה ונוהגין שכל אחד אומר לחבירו לשנה טובה תכתב (טור):