Are you able to supply a bit of
irrigation to the area the weeds went to seed?
If you were to wait a while to let the rest of the seeds drop, you
should be able to force any that would be an issue for fall planting [seeds that don't require cool dormancy] to germinate, start growing and by August turn into
feed for your compost pile.
Plus free cover crops and all the benefits keeping living plants growing in the soil provides.