No, it is not a common expression at all. In normal English usage, we don't come across such a phrase. It seems rather strange and not part of the standard vocabulary or expressions.
Sharing such a story is highly inappropriate as it involves private and inappropriate details about a family member. Family members' privacy should be respected, and this kind of content is not suitable for public discussion.
Well, 'drunk husband loves cock stories' is a rather puzzling statement. 'Drunk husband' is a common enough concept. But 'cock stories' is quite strange. It might be that in some sort of very specific context, like in a group of people with their own unique jargon, it could mean that the drunk man likes stories that are perhaps inappropriate or it could be a completely innocent misnomer for some other type of story. For example, it could be a play on words gone wrong and the real meaning was something like 'clock stories' where the drunk husband likes stories about clocks.
Well, if not in a vulgar sense, 'huge cock' could potentially be a misnomer or a very strange way of referring to something large. Maybe it's a big rooster on the family farm that the stepmom has some sort of story related to. But this is a very odd and uncommonly used way to phrase it if not being inappropriate.
This kind of story involves inappropriate and unethical relationships. Incestuous relationships are not acceptable in society as they violate basic moral and ethical norms. It can cause great harm to family members and disrupt the normal order of family and society.
Share stories that promote family values such as unity, support, and love. For instance, a story about how the family came together during a difficult time. Make sure the story is something that can be told in all family settings without causing embarrassment.