Their tranny flush can loosen sticking components. No, you don't need to remove the tranny or completely remove the tranny flush; as noted, it does not cheat by using solvents.
LUBEGARD Transmission Flush
When you add new fluid, I'd seriously consider their (red) protectant. I did so before I first changed fluid and it definitely made a difference in how smoothly the tranny worked, especially when up to temp.
LUBEGARD Automatic Transmission Fluid Protectant
Double-check the fluid compatibility with your car, but there is no reason their products shouldn't be fine with your fluid.
I'm not a big proponent of doctoring or covering-up things with additives, but considering the alternatives, and their costs or consequences, I would
trust trying something like LubeGard's products; they have a helluva reputation with professionals. If your tranny is already screwed-up, well, it's screwed-up
. But if spending $30-40 for a simple, 2-step fix may correct a minor issue, I'd go for it.