"What is true" does not refer to the entire universe. In "owning up to it doesn't make it worse", it refers to the specific thing "what is true" that you are trying to change your mind about. "Owning up to P doesn't make P worse", because your state of mind is not causally connected to P. In the specific example of finding that X is false:
"X is false" is already true. Owning up to it doesn't make "X is false" worse.
Clearly whatever bad things are brought about by a state of affairs where X is false are already occurring -- because it is in fact false! Changing your mind about X should have no effect on these affairs. Your social situation, on the other hand, is a completely different thing.
The Litany of Gendlin is meant to neutralize fears like "but if god didn't exist, that would be terrible!" resulting in clinging to faith in a god, or "I can't be ill, that's too bad to imagine!" resulting in not going to the doctor.
For example:
Say I live in a bad neighborhood, but I'm kind of clueless and don't really want to believe it. I hear gunshots sometimes, but rationalize that it must just be cars backfiring. I hear my neighbors fighting, but tell myself it must be a TV program that someone has on really loud. I see people hanging around outside, selling who-knows-what, but tell myself that it must just be the local culture, and it's not my place to say that other people can't spend time outside, that's just silly.
The probability of the police breaking my door down because so...
—Eugene Gendlin