Your best option is talk to an attorney. The first consult is often free, and even if it isn't, it will be worth every penny. That's because you'll learn your options. There are ways people can keep all info about them private, but it can require a court order, where you'd have to prove your need for it. Also, you might ask them about a protection order. This can be sent to them without giving any info about you, and it might make them think twice about trying to contact you. If you reported it, they go to jail.
If I were you, I'd try to avoid leaving the country just yet. You'll have a lot going on, and that just adds more dimensions to it. One thing that will make things go easier for you is try to find a group home in your new city. People usually don't require a background or credit check (and you can certainly explain in vague terms why you don't want it). But what's good about these is you can avoid the typical things that get your name online -- utilities, wifi, even rent. Find a check cashing place for when you get your job and cash the check there if they don't pay you with a debit card . You don't need to go completely off the grid, but you just don't want your name out there.
Also, the cops aren't private investigators. They couldn't help your mom find your dad and they won't help her find you. This is another area where the attorney could help, because he'll know the types of things that PI's can access and how to avoid them.
If all else fails and you want to go extreme, just change your legal name. But you still want to talk to the attorney, because this, too, is public record unless you can get it hidden.