For myself, other people's anger makes me really uncomfortable, and I avoid it as much as possible, in part specifically because if I don't, I end up sharing in it, but without a reason or a target. It's really unpleasant because in a sense, it's not real.
Real anger - my own anger - feels complete. Not that it's pleasant or anything - it's still anger. But in a way, it's a sort of relief to feel it, since it at least makes sense. I have a reason for it and a target for it, so it fits. Empathetic anger is weird and unsettling, since it's just there, but it's not a complete, sensible thing.
And you're right about targeted emotions, at least in my experience, and while anger is a good example, it's not the worst.
Grief is awful, because it's such a horrible, desolate feeling, and just that much worse when it doesn't even really mean anything.
Jealousy is another bad one - in fact, thinking about it, I'm tempted to say it's the worst of them all, because it's so unpleasant, and in multiple ways, and it's so entirely pointless without an actual reason or target (it's arguably fairly pointless even with both).
On a somewhat different note, just because I'm thinking about the trials and tribulations of affective empathy - embarrassment is weirdly bad. Partly it's that it's unpleasant, but more it's that it's such a common aspect of other people's enjoyment - there's a great deal of comedy that hinges on laughing at other people's embarrassment, and it's all completely lost on me, because I'm stuck just feeling pointlessly vicariously embarrassed.
Broadly, the way I have to deal with all of it is to try to avoid situations in which I'm going to be subjected to other people's unpleasant emotions, and if I find myself in one, to try to shut myself off from whatever they're feeling. I'm okay up to a point, but I can feel it coming if I'm getting to the point that it's going to suck me in, and pretty much all I can do then is resign myself to it or throw up a barricade and just shut it out. Which sort of ironically makes me come across as aloof - as if I'm insensitive rather than overly sensitive. That gnaws at me, but there really isn't much I can do about it, since I already have enough to deal with with my own emotions, and just don't have the fortitude to deal with everyone else's as well.