When I arrived my first aim was to understand how things were working in a famous university.....
In other words, you are not familiar with the rules (written and unwritten) of US academic culture.
In your case, you joined the PI's team and worked in the PI's lab, got to use the PI's facilities, and so on. The PI has a rather hand's off style, which may be new to you, but it happens here; the opposite is finding a PI who is a micromanager. Most supervisors are in-between, but that is a side issue. But the good news is that the PI was impressed enough with your work to offer you a post doc. So see that as a success on your part, whether you take the post doc or not.
So what about your research and paper in process? You talk to the PI about options for publishing, looking for his/her blessing. It is not something you go off and on your own. Remember, you are on the PI's team and in the PI's lab. Not your own, and don't be confused or misled by the PI's hand's off style. Depending on your field, you may even find that you are second or third author in the paper, the PI being first author. It happens, so don't be shocked. Your goal is to get that paper out, not to fight about who is first author. People in your field will know how the authorship works, so you being second or third author won't be considered second best.