CORPUS CHRISTI, Texas—This Gulf Coast city has ridden a rocket of a rebirth in recent years, propelled by a more than 30-fold increase in oil production at the nearby Eagle Ford Shale since the beginning of 2010.

Corpus Christi’s economy went from stagnant to zooming, growing by 8% in 2012 and 6% in 2013. Workers flush with cash have helped drive average home prices to nearly $200,000 from $150,000 four years earlier.

The...