Many companies are waking up to this kind of problem. They're putting in security measures to help stop it. Two-factor authentication is a good example. It makes it harder for someone to fake your identity.
These measures can also include things like account activity monitoring. You'll be notified whenever someone logs in from a strange computer. It helps you take action more quickly.
Facebook has a bevy of new security tools. You can learn about them here and here. These are things you'll definitely want to enable.
On Facebook, go through and delete all the posts the ex put up. You can do this by going to your Profile page. Hold your mouse over a post. Then click the X that appears in the corner.