What I mean is that they are only independent in the sense that you are free to specify them *independently* as initial conditions. Once you do that all you have to do is take some nice derivatives (no integration constants needed!) and you will have the evolution of your system for all time.
They are only independent in the sense that you need position and velocity to specify the initial state of a system (and thus the further time evolution of the system - you are completely right about this) because F = m a is a 2nd order differential equation and there is an initial velocity...