I regularly have thoughts with the word "shit", examples he/she has shit character, not that shit or I'm sick of this shit, etc. How could I have the thought or say "shit" in a more sophisticated or mature way?
It's a very versatile word: it can describe bad quality ("that movie was shit"), good quality ("that movie was the shit"), general undifferentiated possessions ("that's my shit"), narcotics ("I smoked some shit"), anything unwelcome ("I can't eat this shit", "He was talking shit about me", "The little shit scammed me again"), general expletive ("shit, the boss is coming"), and of course the literal excrement.
All of them have alternatives; but there is no single expression that would cover all of them. "He has a bad character", "Not this thing", "I am sick of this situation".
As Andrew says, these are all very generic; in a specific context, there might be much better choices. For example, "He has an odious personality", "I can't handle her histrionics about the holiday trip again", "He is driving me insane with his constant criticism".
Of course it depends on the details of the particular situation, so these are only a few examples.
he/she has shit character.
He is remarkably untrustworthy, undependable, unreliable, and I have never met someone who I disliked as much.
not that shit.
Not this absurd nonsense, yet again.
I'm sick of this shit.
I am extremely tired of your behavior.
or, in an affected British accent, lol :
I have grown weary of your temper tantrums.