I don't have good answers for you, but now I know the name of the weed that's infested parts of my lawn and some of my beds! Thanks!

Honestly I've had very poor luck controlling with herbicide. It requires repeat applications over a few seasons to significantly reduce an established infestation. I've used both Roundup (non-lawn areas) and Weed B Gone Max (lawn areas) with limited success.

In beds I have found manual control (hand-pulling) much more successful. It takes diligence. You need to pull in the correct conditions and do so frequently. With shallow-rooted rhizome plants, when you pull don't think of pulling up, think of pulling sideways. Find the densest cluster and that's where your main root mass probably is. Start from the edges and pull towards the direction of that root mass.

Work when the soil is loosest. That would be after rain for heavy soils, and after a dry spell for loose soils. Again, start from the edge and work towards the center. While you are gaining control, pull any flowered plants even if they break and you know they'll grow back. Pull slowly and carefully to try to get as much root as possible. Any remaining roots will resprout, so what you want to do is each day where you have time to weed, go back over the spots you've already done before working in towards the worst areas. The goal is to kill it off from the outside in by getting any new plants from runners (which might be quite far from the main plant). This prevents the patch from getting bigger while you're working in exhausting the main plant.

I did use spot Roundup after lots of hand-pulling. I've had great success with this method in beds with both sheep sorrel and mugwort. The key is to work on it at least a couple times a week and be diligent about it, always working from the lightly infested areas in towards the center and never letting the new offshoots get the chance to establish deeper, thicker roots.

You might try a pre-emergent to keep new seeds from sprouting, but I haven't researched their effectiveness against this weed.

I have no help for you...I just find it funny that some of us try to eradicate a plant while others try to help it grow. I was disappointed when my in-laws decided to do us a favor and mow the lawn, cutting down all of the sheep sorrel in the process. I loved eating the sour leaves while I strolled through the garden.