Not a teacher, but a British native.
A person of good breeding, especially a lady, should not use "I don't give a sh*t" in public.
A man should also not say this in the comany of ladies. It is likely to cause offence.
"I don't give a damn" is much less offensive nowadays, and I doubt if anyone would 'bat an eyelid' if a lady were to use this phrase in public. They probably wouldn't give a damn, in fact!
If you want to be really sure that no offence is taken, I suggest "I couldn't care less".