How do I cope with clients leaning on me on their worst days?

You can shepherd anyone through a claim, calm them, guide them, get them through. And that means people come to you at their lowest, after the fire, the crash, the loss, and lean their fear on you. Doing that well, over and over, wears on you. Absorbing other people's worst days is real labor, even when it does not show. What makes you good at this, your steadiness and care, is also what drains you, because you take some of their weight each time. That is not a flaw to fix. It is the cost of doing this work with a real heart. But a cost unmanaged becomes burnout, and then you are no use to them or yourself. You have to tend the one doing the tending. So build small ways to set the weight down between clients. A moment to breathe before the next call, a hard line between work and home, one person you can offload to. And remember what you are actually giving: on the worst day of someone's life, you were the calm they held onto. That matters enormously. Just do not forget to leave some of that calm for yourself.