A common theme with Plastic Man is him having a kid, whether adopted or biological. So it wouldn’t be totally illogical to assume that the Plas of Justice League Action has a kid stashed away somewhere.
Through several cartoonishly contrived coincidences, a burglars ill-gotten loot winds up in Plastic Man’s bed as he snoozes. Upon waking up, Plastic Man assumes he’s been committing crimes in his sleep.
Another thing about that Plastic Man pilot, how did he get arrested in the first place? Was he arrested right after the chemical incident? Did he turn himself in willingly? Either way, putting Plastic Man in a jail cell probably would have worked as well putting the Human Puddle in a jail cell.
If the show had taken off and there was never a gag of Plas removing his hands from cuffs or stepping out of a locked cell, I’d have been disappointed.