I have full custody of my child, but he would like to live with his father. Now when I give custody over, do I pay child support?

Yes and maybe. When custody is changed from one parent to the other, the child support agreement is modified, The father will of course ask for his payments to stop, but he may or may not ask for support.

And the amounts will be based on incomes so it would not be the same amount of course.

I would ask, why would you think that you should not pay ?