This is a good question, and something I am personally probably still struggling with. I feel like the greatest progress I have made on this is focusing more on listening, and less on conversing or having answers. People generally appreciate if you take their concerns seriously, but are easily offended if you offer obvious advice that brushes their concerns aside.
> How to be more empathetic?
According to some research [1], reading literary fiction has a greatly positive impact on empathy and the capacity to process emotions. I have even heard the argument that the boom in novel production in 18th century England created a cultural shift, as the increased capacity for empathy made people more morally conscious on a variety of issues.
As I believe that giving a somewhat empirically grounded answer is very much in line with the spirit of this forum, I shall therefore suggest everyone to read more novels.
> What common failure modes do you hit in your relationships as a low empathy person?
I don't think that having low empathy has ever been a "failure mode" to me -- but regardless, I feel like I have emotionally matured in a way that I wouldn't describe myself as "low empathy" anymore. It's certainly an issue when trying to pick up girls, though. (Wouldn't it be so nice if you just knew people's intentions?)
> How do you avoid them?
The crucial thing is that when you have a conversation, you should try to be mindful of what the other person is looking for, even on an emotional and possibly unconscious level, and try to accomodate that first and foremost. This always involves some guesswork, but it is worth it to try and make an effort. As noted elsewhere in this thread, often people just want to talk for support and are not looking to solve a problem. This can cause some friction with math-sy people, who often have a strong urge to start running solutionizing when encountering an unbalanced equation.
Taking my own advice here, I also want to tell you, OP, that it's not really an issue to not be great at those things. This is by no means limited to technical people, and I actually believe that most people struggle with this sometimes: there's the marketing manager who will stop you at the water cooler and talk at you for 10 minutes about the amazing new project she's doing, with you awkwardly trying to exit the conversation; and there's the menial worker who feels awkward around "smart" people, often avoiding them altogether, as if he felt they could bite him; and so on. In the end, we're all just human, and the important thing is that we're here to learn to respect and understand each other. You seem to be on that journey, and that is the important part.
[1] https://www.scientificamerican.com/article/novel-finding-rea...