Hello KDTsai,
It sounds like you'd like to create accounts for your children and set up Family Sharing.
To use Family Sharing, since you're the family organizer, you'll need to have a payment method on file in the iTunes & App Stores. Check out the "Start a family group" section of this article: Set up Family Sharing
You also mentioned that the verification code you're receiving doesn't work. Are you having trouble signing in with your Apple ID and using two-factor authentication? If so check out this article: Get a verification code and sign in with two-factor authentication
If you're experiencing something else, please provide a few more details.
Have a good day.