In the Dallas-Fort Worth area, 24-hour totals ranged from just under 3 inches to more than 5, the weather service said. Flood warnings were posted in many areas.
A number of drivers had to be rescued after their cars stalled on flooded streets, WFAA-TV reported. Police warned drivers to stay out of high-water areas.
Austin was under a tornado watch, KVUE-TV said. Firefighters responded to one possible twister, and Battalion Chief Palmer Buck said at least three buildings were damaged, including one that collapsed and one where windows were blown out.
As rain tapered off and winds dropped in the state's northern region, the weather service posted a flood warning for the Corpus Christi area. Forecasters predicted there will be flooding along the Guadalupe River this week.
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