Godsworn characters are those who swear only to serve the Gods - all the Gods, making service to the gods their primary calling in life and putting all other concerns secondary. This is different from being a Disciple; a Disciple is a member of the laity who will still go about their normal day activities, they just also have duties as part of a temple also, and are usually connected to a single God as a vocational order. A Godsworn is one who is sworn to the Pantheon (and the crown) and does not take other oaths that could ever interfere with that calling.
Godsworn do not marry, own land, have children, pass inheritances, or have any obligations that could interfere in their service to the gods. They hold the one oath they have given to the Pantheon to be above all others, and this means they do not give separate oaths to a spouse, and do not take on the responsibilities of a family. They give up their familial ties when they take their oaths. Some families will cut off all contact, others keep informal ties - how a family and a Godsworn individual handle it is up to them, with the understanding that the Faith takes precedence in all things. It is a lifelong commitment that can only be released by the leadership of the Faith, and the laws and rules are enforced upon the faith by the faith.
This does not preclude them from having affairs, friendships, or long-standing relationships. It just means they do not marry, and do not have a family. The Faith does not accept Godsworn with familial relationships already, except in cases where, for example, a parent of grown children with no spouse wants to take vows. Then the familial obligations are considered fulfilled, and nothing stops them from making their vows to the Gods. In short, service to the Faith is a privilege, not an escape for those wishing to abrogate their responsibilities. Likewise a member of the Faith who makes conflicting oaths or who has children is considered an oathbreaker, and consequences ensue based on the severity of the situation. Anything from explusion and excommunication, to punishment as one of the Silent Reflections is possible.
Duels and other matters of honor are not often fought as Godsworn members, save in cases where the honor of the Faith is in question. For smaller matters, as with minor insults and other follies, they would most likely decline a duel or hire a Champion if they absolutely must. This is partially owing to dueling for silly reasons not being serious, but also because when a member of the Godsworn is involved, it takes on an entirely new level of seriousness that might not otherwise be part of a duel between two members of the nobility.
Other day to day slice of life activities are fine. Godsworn can have personal belongings - maybe they like a nice wardrobe, or they like collecting art, or whatever else they like. They can make friends and do everything everyone else does. Some will go to war, some go on smaller missions. Some are diplomats, some preach sermons. The only thing they cannot do is make oaths and take on familial-type responsibilities - their oath to the Gods supercedes all of them.