I suggest that one of the blessings of the weekday Amidah (page 120) covers confession without the need to say Tachanun:
Forgive us, O our Father, for we have sinned; pardon us, O our
King, for we have transgressed; for thou dost pardon and forgive.
Blessed art thou, O L-rd, who art gracious, and dost abundantly
This covers the general need for confession but one can mention an individual sin in the prayer if needed, see this article.
And then see especially O Ch 118 (1) MB  who says,
טוב להתודות בש"ת ויאמר חטאתי עויתי פשעתי
It is good to confess in
the blessing of “Shomeia Tefillo” and say, “I have erred, I have
sinned, I have transgressed.”
CONFESSION ON SHABBOS
Din.org.il says that one does not say vidui on Shabbos.
Rav Avrohom Yosef at shut.moreshet.co.il says
מותר לחזור בתשובה בשבת ! אמירת וידוי בשבת הוא מעשה לא נכון כיון שהוא
מביא לידי בכי שאסור.
One may do teshuvo on Shabbos! However vidui is not appropriate
because it brings to crying which is not allowed on Shabbos.
The question is how does one manage to do teshuvo without vidui. I may suggest
(1) to do as much of the process of repentance as possible and leave the vidui over until after Shabbos or
(2) that one confesses in thought but not in words (not ideal) and repeat after Shabbos.
The Jewish Encyclopedia Daas writes about death-bed confession (allowed but not in front of women and children in case it upsets them too much.) and confession when fasting on Shabbos after a bad dream (allowed).
CONFESSION ON SHABBOS 2nd instalment
Amongst the entries in כמוצא שלל רב, I found the following question asked:
The Gemoro in Kiddushin 49b says
על מנת שאני צדיק אפילו רשע גמור מקודשת שמא הרהר תשובה בדעתו
If someone even a completely evil person proposes marriage to a woman
on condition that he is righteous, she becomes his wife in case he has
had thoughts of repentance.
He does not seem to have performed confession – how can he be a righteous person?
The author of “קרית ספר" that confession does not prevent repentance. However, the repentant person still has a positive mitzva to perform.
The Minchas Chinuch holds that confession is essential, like the Rambam and that repentance wthout confession cannot atone for the past but is good enough to make a fit person for the future.
Other answers are available there.
So it seems that according to some, repentance can be performed on Shabbos.