Texas patients are rushing to get abortions before the state’s six-week limit. Clinics are struggling to keep up.

When the woman started crying in the ultrasound room, Joe Nelson tried to comfort her, as he has comforted dozens of other patients who are too far along to get an abortion in Texas.

She was a single mother with two kids at home, experiencing a rare pregnancy condition that had left her too nauseous to work, said Nelson, a doctor at Whole Woman’s Health, an abortion clinic in Austin. The woman was over the legal limit established by Texas’s restrictive new law, Nelson said, but just barely. A few days...