I'm going to continue to be friends with someone who falsely accused my closer friend of something serious
How can you comfortably still be friends with them?
It's just that she was in a bad position in the situation, I understand everyone has their problems.
This is a sticky situation. If the accusations are so bad, then maybe ditching both of them is the way to go here or you can try to mediate between them to resolve their issues.
I would do anything for her, I promised that
It's great you want to still be friends, but it would be healthier to break off the relationships now before anything worse happens. Perhaps you can try to console both of them and make them feel better?