MacSwiss, what you need to do is give each family member a different Email. Go into - "Settings" , "Messages", "Receive At", and add a different Email for each user to their iPhone. Do this for each family member.
Example - Your iPhone - Dad@me.com, Your wife's iPhone - Mom@me.com, Child 1 iPhone - Kid1@me.com ,and Child 2 iPhone -Kid2@me.com. If each family member already has their own Email account then you can use those Email address. No need to create any new ones.
Once you have added a new Email to each device delete the original Email that was there which should have been the same as your Apple ID. Also make sure the Email you are using for each family member is also under their name in the "Contacts" list.