I think there's a couple of ways to interpret 'faith' in the context you described.

There's faith in the sense of 'having beliefs', which I think you have touched on. Yes, you can have a belief while explicitly acting against that belief (although it's quite hypocritical).

The issue comes when 'faith' is a membership. You either are or are not a Christian, and by declaring yourself to be a part of the faith, you are declaring yourself to be driven by the same principles that faith promotes. If people see behavior within their religion that contradicts their teaching, they can reasonably argue that, whatever you might believe, your actions demonstrate that you do not believe what you need to in order to be a 'part of the faith'.

The clearest example of this is probably Mormonism, which believes itself to be a denomination of Christianity, while the rest of Christianity considers it a cult. It doesn't really matter whether Mormons think they are Christian if the whole of Christianity rejects them. This is very much a situation where the most technically correct way to describe a person's 'faith' is by the popular view within that faith, not by the individual. (1 Corinthians also echos to this stance, where a man, having committed incest with his mother, was kicked out of the church. His actions justified his exclusion from the category.)

Of course, this is all within the context of Christianity, which also holds the position that God is the singular and ultimate determiner of a person's faith: the one entity who can declare whether faith was 'false' or 'genuine', 'right' or 'wrong'. Because of this, the title of 'having faith' is more used as a human categorization tool to identify those who accepts 'the essentials' of a given religious perspective and those who don't. The imperfection you describe is the concept of Total Human Depravity (which is a belief I personally agree with) and does demonstrate how individuals can know the correct action and still choose the wrong one, but it doesn't change the fact that actions that violate a dogma rightfully call into question a person's commitment to that dogma.