I have a couple, and I hate them! I can't remove the two that are in my immediate yard. The roots pop up all over the place, and as you found, if you cut them, they just continue to grow. I've cut them and put poison ivy brush killer on it and it won't grow there, but anywhere else, new growth pops out.
I just looked this up, it may be helpful:
Tell me how to kill sweet gum tree roots!! | Carolina Shooters Club
Wish you luck. I may try the above on what stumps are still sending up shooters!