Confusion about the meaning of steady current

I am trying to learn some elementary EM, but I have some confusion about the basic concepts of steady current.

Suppose I have a wire of uniform cross section area. The current is always flowing from left to right.

I imagine that I can cut a segment of this wire (with the area vectors at both ends parallel to the flow of current) and compute the surface integral $intint_S mathbf{J}cdot dmathbf{A}$, where $mathbf{J}$ is the current density vector, $mathbf{A}$ is the area vector, and $S$ is the boundary of the segment. I assume that $mathbf{J}$ does not vary with time.

I believe that the magnitude of $mathbf{J}$ can depend on its position, so lets say that its magnitude is greater on the right end, as compared to the left end. Because of that, the surface integral $intint_S mathbf{J}cdot dmathbf{A}$ should have a non-zero value (the dot products on both ends do not cancel).

However, $intint_S mathbf{J}cdot dmathbf{A}$ is precisely the net change in charge out of this segment, i.e., $-frac{dq}{dt}$.

Now, for a steady current, $frac{partial rho}{partial t}$ is zero at every point, thus the $-frac{dq}{dt}$ should also be zero, which contradicts my understanding that $intint_S mathbf{J}cdot dmathbf{A}$ can be non-zero.

This is perhaps a very stupid question, but I just cannot figure out what has gone wrong. Any help would be greatly appreciated~