Should I avoid a person if being near them sends me into panic attacks (I'll see them then find myself bursting into tears, rocking myself back and forth, and repeating, "I'm not safe")?

Feeling unsafe and fearful around others is something that can be quite difficult to cope with, especially when you’re alone. It sounds like there could be trauma associated with this person and in that case, something you could try is to keep your distance on the regular if it’s possible, for the sake of your mental and physical well-being. When you see this person, it sounds like your body is flung into survival mode and what follows that is crying and self-soothing behavior. I can only imagine how helpless being around this person would make you feel, and I’m sorry it’s like that. It can help to tell yourself (as long as it’s appropriate and feels right to do so — if you aren’t physically in danger) that you are safe, and you’re going to be okay. I am not aware of the fuller context, but I hope you’ve been able to manage the panic and find safer people and situations.