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The specific situation of the "sorry for existing" loop you mention is a common overlap with Apologises a Lot- it can be sincere, a passive-aggressive effort to drag everyone down with them, or some ambiguous mixture.
The Killjoy is similar and can overlap, but alone they tend to favor a more actively annoying sort of whining. The Bore is another potential overlap, when they're not only sad but sad in an uninteresting way.
Edited by Scorpion451

A character whose guilt and depression causes other people to be annoyed, which makes the character feel guilty and depressed, causing other people to be annoyed, causing the character to feel guilty and depressed, which [etc.].
Either side can be portrayed as in the right: either the character is The Woobie surrounded by All of the Other Reindeer, or the character is a whiny self-centered Manchild who needs to stop killing the mood and the other people are right to cut the character out of their lives.