How do I deal with it?

Well after I'm able to calm down and think, instead of just feel like an exposed nerve...

I convince myself that its the perfect opportunity for "exposure therapy" and that I need to run with it.

What that means for me is this;

Not everyone can meet all my needs all of the time. Sometimes the person is drowning in their own bullshyt, sometimes I'm just too damn much. Whatever the case may be, my objective is to actively proceed WITHOUT them holding my hand, and thus not only be like "see Lia you didn't die without them!" but also be proud of myself for handling it like a boss.

Sooo that's how I deal with someone being insensitive.

But, that being said, there are different variables to consider...

If the same person is routinely behaving in the same insensitive manner, and are just plain selfish, then they have to go.
No ifs ands or buts.

In the case of my bf? He seemed emotionally distant when we got back together, and I felt slighted. But the truth is, he was still recovering from the confusion of me emotionally and mentally drowning him and he understandably had his guard up.
All he needed was some time and education though.

Once I came clean about the BPD he relaxed quite a bit as I suddenly made more sense than Bipolar ever explained...

Your cousin seems a bit tricky since she's not insensitive when you're feeling stable. Maybe she doesn't know how to handle emotional crisis, or maybe she actively pulls away in order to prevent being your crutch?... idk

Like I said, there are quite a few things to take into context, since people are complex, so I would talk to her about it before deciding on how to proceed.