From what I've seen, it didn't get really bad until about two weeks before he died with Filipe. There were little moments, like his taking on the Joker and kicking him in the balls, but on the whole, I feel like they tried to play it up as much as possible so he could be killed and it seemed fine. I also note that Bruce never believes him about Garzonas while he's alive. It all feels as if it's supposed to make you believe this 14 uear old boy was a violent killer and has never been anything else, which just isn't true.
Damian has the potential to get better without the same psychological scarring because it's more learned behaviour and he is only ten. I can see why they think it'll be easier to change him, but if they use that to show how bad Jason was all along (!) I shall not be amused.
I must wonder how much of the problems is Jason wondering that every time he loves someone, they die too? Both his mothers and his fathers have died. In that respect, Bruce and Jason are very similar. I think Jason is battling that line of thought with thinking that Bruce didn't love him, because all he ever saw when he looked at Jason, was Bruce himselfs (or Batman's, I suppose) worst qualities and his worst mistake. That is a lot of weight for a 19/20 year old to carry, never mind a 13 year old.
It could also be that he's sick of the people that do love him dying, so it's easier to be angry than to be hurt, because then he's remembered for being a lethal vigilante, not for being a victim and a mistake.