It’s nice to meet you, @Audrey_Mehring - thanks for reaching out! I’m the Community Manager at Asana.
Many of our creative clients use Asana for content calendars. I suggest that you visit this thread about content calendars in Asana and also search “template” in the search bar here in the Community. There are a few examples of templates and summary projects here that I think will help you feel more organized with Asana. You might also enjoy this video about how Autodesk uses Asana for editorial calendars.
Regarding personal responsibility, this is a very good question. The Asana Guide has helpful recommendations for getting your team started with Asana. Walking through the recommended flow can be very helpful. I also recommend actively establishing Asana conventions on your team.
Please let us know if any of these steps are helpful.