On a warm day in May 2018, Ashton and Charmayne Thomas swiftly booked a flight from Arizona to Houston. They dropped off their two eldest children, Emmett and Adelyn, then 4 and 3, with family boarded a plane to Houston with their 9-month son, Wyatt.

“I want to see Ashton tomorrow,” sarcoma specialist Dejka Araujo, M.D., had told Charmayne on the phone the day before.

Doctors in Arizona had found a 10-centimeter mass in...