Did you do the things in this support article? --> What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support
Your description of the flow of things is not fully consistent with how Apple IDs work on devices. Yes, if you did not erase your device first then your information may have combined with his on the old iPhone, but it shouldn't have transferred over to your new iPhone unless your son was using your old iPhone wile it was still signed into your ID. If that's the case, make an ID for your son, erase the phone as in the previous link I provided, have him sign into the old phone with his new ID, and then on your account you will have to delete the information he entered.
Make sure that each family member has a unique Apple ID --> Make sure that each family member has a unique Apple ID - Apple Support
For a child: Create an Apple ID for your child - Create an Apple ID for your child - Apple Support
Click here to see how to 'Create a new Apple ID' = create an Apple Account. --> How to create a new Apple ID - Apple Support