As the war went on, there was a need to pull in more men and a lot of guys who had been exempt earlier were called up. My grandfather had two kids, but he was ordered to report for induction. It was only after he’d gone through the whole process that they pulled out a letter from his employer saying he was doing necessary war work and sent him home. But that was the basis of his exemption, not the two kids.
Yes, during the last year of the war the draft boards were running out of draft-eligible men under the existing exemption rules. So, older men and men with children became eligible for the draft.