I added PID control for the current to the program. I used the standard NI PID routine that comes with the FPGA toolkit. All I have to do is feed in my PID coefficients, setpoint and control variable and it takes care of the rest. I just added it to the output of my current measurement routine so it updates the PWM value once per cycle. I also added a switch so I can change between PWM control and current control.
The initial tests are promising. I made the setpoint alternate between small positive and negative currents and watched the current values as it cycled. I managed to come up with PID coefficients that quickly reach the setpoint values without going unstable. The values were small enough that nothing moved. There is still a lot of testing to go, but the PID control of current looks pretty good at first glance.