We do not say “Mizmor LeSodoh” on those days when a Korban Todah could not be brought.
So we do not say it on Shabbos or Yom Tov because you could not bring individual offerings on those days and we do not say it on Erev Pesach or Chol HaMoed Pesach because it contains Chometz. Nor do we say it on Yom Kippur and Erev Yom Kippur.
The blessing for deliverance from danger (Birkas Hagomel) was instituted, according to many authorities instead of the Korban Todah ברכת הגומל במקום קרבן תודה איתקן
and see related birkas hagomel: based on the toda?
Can the Birkas Hagomel be said when the Korban Todah could not be brought? I have certainly heard it on Shabbos!