Sounds dreadful. I know here (and I'm just a res block) they are one of the few things that survived where nothing else did, but they absolutely took off in areas that got water.
There are bindii sprays of course, designed to be used in the growing period. You might try your state DPI to see if they can recommend anything for a larger scale. Of course, chemicals may not be feasible or your preferred choice given your area/water catchment and what you ultimately want to do with the land. Your problem is probably going to be worsened by years of seeding that may just lie dormant and pop up again when conditions are right - like when you have turned over or improved the soil. I know there are seed inhibitors but same applies - whether its safe to use them or you want them.
As far as reducing the problem act soon as winter is the growth period and you want to stop more seeds. Is it feasible to lay black plastic and bake them out then plant through holes where you want things to go?