Family Sharing iTunes Account
Can (adults) use their existing credit and their own credit card, or must every purchase go through the family organiser ?
No, only the family organiser's card can be charged, not the cards of members within the family - if the family member has a balance on their account then they can use that (so they could buy gift cards, if they are available in your country, and use them instead) : Family purchases and payments - Apple Support
After you set up your family, any time a family member initiates a new purchase it will be billed directly to your account, unless that family member has gift or store credit. First, their store credit will be used to pay the partial or total bill. The remainder will bill to the family organizer's card. As the family organizer, any receipts generated by the transaction will be sent to you.