DC motor model issue
\$J_{load}\$ is the moment of inertia of the load; mathematically it's the equivalent of mass in a system where you're dealing with linear motion.  Here's the details: $$T = J \alpha \\ F = m a$$ where \$T\$ is torque \$J\$ is moment of inertia, \$\alpha\$ is rotational acceleration (\$\ddot \theta\$), \$F\$ is force, \$m\$ is mass and \$a\$ is linear acceleration.

Increasing \$J_{load}\$ will slow the response of the system to changes in voltage, because it takes more torque to spin the motor up.  But \$J_{load}\$ does not change the amount of torque used by the system once it is at speed.  That is why when it eventually settles, it does so to it's previous speed and current.

This is quite apparent in the math, if you can pull your head out of dependence on Matlab for long enough to spend some time with a pencil and paper.  I highly recommend you do so; the equations of motion are first order and linear, so should be easy to solve.