Dafür haben (or tragen) die Krankenpfleger in einem Pflegeheim täglich die Verantwortung.
is right. See also the examples in DWDS.
If you want to have an explanation for the correction, you need to ask the people who told you to use darüber. I don't see any reason and for my native German ears dafür is perfectly fine, darüber sounds strange.
The word Schuldi exists
and I think it's what you've heard.
I've heard it sometimes in conversations myself, there is also an entry in Mundmische (the German version of the Urban Dictionary).
It's an exclusively colloquial, exceedingly informal and very, very 'cute' (some would say annoying) way to say Entschuldigung. Caution: may appear ridiculous if used ...