Hundreds of people who say they suffered physical or sexual abuse at two state-run reform schools in Florida are in line to receive tens of thousands of dollars in restitution from the state, after Florida lawmakers formally apologized for the horrors they endured as children more than 50 years ago.

At its peak in the Jim Crow 1960s, 500 boys were housed at what is now known as the Dozier School for Boys, most of them for minor offenses such as petty theft, truancy or running away from…