It is still possible that your father's story was correct. Since he registered the births late and all at once, it may have been simpler for him to say they were all born in Zuromin.
It is also possible that the family lived in Radzanow at the time but that the mother went to Zuromin to the home of her mother or something like that to have the baby born. My husband's family lived in Lwow, but his mother was born in Krakow simply because that is where her mother went to be with her mother at the time of the birth.
