I think that apologizing excessively is sometimes also a cry for attention as it puts the person who apologizes in the center. Maybe instead of focusing on the self and perceived wrongdoings, talk about thoughts and ideas how to improve the situation in a factual, more objective manner. That might actually lead to something constructive (proposing improvements and solutions) instead of simply just acknowledging that something is not right and debating whose fault it is. This strategy won't work in every context but I think can be used in many.

The other good tactic is to wait and think before saying something driven by a momentary feeling or impulse. Just like when we discuss things about the therapist-client relationship here on PC, it can become more obvious upon reflection that it's not our fault or not entirely. I think too much apologizing is also often an attempt to please someone else when we don't know what else to do. Again, focusing on the facts instead of persons and blame might help some.