Thank you for writing to Just Answer.
I think your conclusion is correct: it doesn't sound like she has fully healed from her prior relationship.
My advice would be for her to see a therapist to work through the issues. Leaving a six year relationship and getting involved only two months later didn't give her enough time to process the lessons and hurt from that relationship.
As you noted, her ex found someone else very quickly, and that might have caused your friend to want to find someone quickly also, to comfort the hurt she was feeling.
It would be possible for her to work through those old issues in therapy so they don't interfere with her ability to form a healthy relationship with you. You do have to accept the possibility that even though she cares for you, this might have been a "rebound" relationship, based on need rather than on true love. I hope for your sake that this isn't the case, but you need to be aware that it's possible.
Encourage her to speak with a therapist and heal the old wounds so she can eventually move forward.