At first, Richard “Ricky” Harveston couldn’t swallow his hamburger bites. Then, he struggled to drink his milk, and what he managed to get down came right back up.

Ricky’s local Orange, Texas, doctor said that he had acid reflux. “But in the back of my mind, I knew what wrong with me,” Ricky says. His dad died from esophageal cancer at age 74, and his half-brother died of the disease at age 46. Like Ricky, both were heavy smokers...