Velcro Weed? What is the name of the spindly weed with small blue flowers? Live in East Texas? Anybody know.

We have a small garden area that is constantly overrun with what I call "velcro weed" because it sticks to itself and my clothing. It's the only thing in the garden to which I am allergic (have to wear gloves when pulling it). It is very thin, with long runners and has very small purplish-blue flowers. It covers everything once it's started and smothers the plants. What is it and how do I get rid of it, short of digging up the entire garden and/or spraying it with weed killer?

It has very shallow roots, fortunately, so I've been pulling it out of the lawn and garden beds, but as I said, I am apparently allergic to it, so I'd rather get rid of it using some other method, if possible.