Are there AA meetings you can attend in the area? There are online ones too:
Online meeting resources
Attend one in the morning, one at lunch and/or later in the day. Sounds crazy? Maybe. But is it any crazier than getting wasted? And there’s no hangover.
You mention your family’s drinking culture. Lordy lord that is a pain; so many people and places are pickled in the drinking culture.
You don’t have to spend more time there than you want to. Your goal is to see your father. You can do that, and if you want, stay in a hotel, or with a friend, or you can just go for a couple days and not a week, or _____.
Get creative and pull out all the stops. The basic truth is this: you’re allowed to do anything safe and legal to maintain your sobriety.
You may also be worrying about family saying, Hey why don’t you have one with us? You don’t have to do that. If you need to leave sooner, then leave; or you can just walk around with a cup filled with Perrier or diet root beer. Whatever. If there’s something already in your hands, no one will ask.
Take care brother. You are a good person who deserves a safe, sober life.