AUSTIN, Texas—U.S. Immigration and Customs Enforcement was prepared to release three men convicted of sex offenses against children, Texas officials said, in an apparent misapplication by authorities of enforcement directives from President Joe Biden’s administration. The three were not released after discussions in recent weeks between the state prison system and immigration authorities. But the process of keeping them in custody raised alarms that ICE was declining to detain convicts contrary to immigration law, officials said. ICE has dropped “detainer” requests against 26 people in Texas in recent weeks, said Jason Clark, chief of staff at the Texas Department of Criminal Justice. Immigration authorities issue detainers to local or state law enforcement agencies holding a person who is potentially in the U.S. illegally. When a person in the U.S. illegally completes their sentence for a crime, ICE can seek to take them into immigration custody. In previous years, including under …