Short answer: Yes.
“I care not” is idiomatic, but usually only as a standalone phrase. It has a slightly archaic feel to it, which makes it sound formal, which again makes it sound humorous. It is very common to use words or phrases from an inappropriate register to convey humour.
It is not uncommon to hear something like this:
They all say my new couch is horrible and the colour clashes with everything else in the room. I care not. I love it, and nobody's going to make me get rid of it!
Using the slightly archaic/formal pattern here, the speaker conveys a slight condescension toward all those people who dare to criticise his/her new couch, but also frames this slight condescension in a bit of humour so as not to be downright rude.
However, when care is transitive, this modal pattern is not common; the following is a lot more unusual:
They all say my new couch is horrible and the colour clashes with everything else in the room. But I care not what they say about it. I love it, and nobody's going to make me get rid of it!
It's not impossible, but it's even more highly marked, to the point that the speaker just ends up sounding odd, rather than using a hint of archaism and formality to add a nuance to the words spoken.