Interesting plant, Steve. I have quite a bit of it in my "wildflower" garden. I assumed the seeds had been part of my package- perhaps not... I do like to imagine women carding wool with those seed heads. I was thinking your wife would enjoy them for wreath making- but they are sharp!! If she ever renews her interest, let me know and I'll send you an unlimited supply!
This isn't a guess, but I know how the OP feels about the plant. I've beed trying to rid my garden of sweetpeas for 3 yrs. now. They are very invasive to my garden.
I found out last summer that it's not so bad if you keep cutting them down, and they'll just adapt into a 'groundcover', sorta like Wild Buttercups and Clover do in lawns. Unfortunately, the SPs won't flower then, but they won't die either. If I let them live, they swamp everything, even the peonies! There have been a few plants that I don't have anymore because the SPs grow faster than they do...
The other problem is that they are growing out of some cleavages in the bedrock, I think there's about 5 'bushes' in 3 spots; I've tried uprooting them. It's amazing. It's like there's little cracks in the rocks desinged especially for the roots of those plants to sink deep into...