There are differing customs as to whether or not Abinu Malkenu may be said on Shabbath. This applies to Shabbath Shuba as well as the times Shabbath falls on Rosh Hashanah or Yom Kippur.
The Ashkenazi custom, in accordance with the Rama, is that Abinu Malkenu is not said on Shabbath. There are differing customs among Sephardim. Some Sephardim say Abinu Malkenu on Shabbath, whereas others do not. Those who do say it must be careful not to say the portions of forgiveness and sin, such as "Selah Lanu" (forgive us) or "Hatanu Lefanekha" (we have sinned before You), on Rosh Hashanah and Shabbath Shuba.
The instruction that I give at Midrash BEN ISH HAI is that Abinu Malkenu must not be said at all on Shabbath, whether on Shabbath Shuba, Rosh Hashanah that falls on Shabbath, or Yom Kippur that falls on Shabbath.
(See Sh. 'A. 626:3. Rama 626:3, 519:3, 602:1. Kaf Hahayim 584, Oth 8)
