I think the phrase "go exercise" is spoken in colloquial English. But I can still find the phrase "go exercising," even in Google books. Like the excerpt below:
I like to exercise, but if I'm stressed and I want to exercise, it's different than it used to be," she says. "Before, if I wanted to go exercising, it was out of anxiousness. It was not relaxing; it just made me worse. I think it's good to exercise, because the body needs to exercise, but not obsessively, and not perhaps in that moment when you are very stressed and anxious, like you're ready to run a marathon. For me, it's just walking with my dog. I do that a lot. And then I feel calm and relaxed.
Is this an American/British thing? Or is it plain wrong?