My mother always takes her problems out on me and doesn't listen to me.

If you move out, your mother could report you as a runaway and have you brought back home.

And the family member could be charged with parental interference.

So no you can not legally move out, If your dad is not with your mom, and involved in your life, he could try and go to court to get custody changed to him.

But as noted, you only get in trouble and get the person who takes you in, into trouble