"Connected to my personal account" is something I often see people talk about, but it just makes me think that they're using Facebook wrong somehow, or using terms in a way I don't understand them.
Facebook only has "personal" accounts. You have your identity, and that's that. You can't connect to Facebook in any other way, because you're a person and you have an account.
SFC connects your website to Facebook using what FB calls an "Application". This is really just a name for the point of connectivity.
Now, that Application is owned by somebody, or multiple somebody's. All those owners are people, and they have personal accounts. But again, the App isn't important, it's the connection point only.
You may have a Fan Page for your non-personal-brand. This may be for the website, or the church, or the entity-of-your-choice, but the Fan page too is owned by a person or some multi-person-set. SFC supports posting to Fan Pages as well as to people's individual timelines.
Each module in SFC does something different. The Login module lets people log into their WP account with FB credentials. The Publish modle lets people cross-post their Posts onto the FB Profile or Fan Pages. The Comments module lets people leave comments using their FB credentials.
So, I'm not certain what exactly you mean when you say that you don't want it connected to your account. It has to connect to somebody's account, and "personal" accounts are the only kind of accounts that exist.