Here is the solution from another user:
hanks to Milo's and MathSmath ideas, I was able to figure out a solution myself. Apparently a lot of people have issues with W3 Cache and sharing on Facebook. There's quite a simple solution to it, though!
Go to the W3 Cache plugin settings, more specifically to "page cache" and make sure to reject these user agents:
facebookexternalhit
FacebookExternalHit/1.1
FacebookExternalHit/1.0
This will block W3 Total Cache from sending cached pages to Facebook.
Personally I still have issues when using the "like" or "share" button in my blog. The "Like" button sometimes doesn't provide the right thumbnail, but I'm sure I can fix that, too. I might update this answer then. The "share" button I use doesn't work at all. I guess I need to find another code for the Facebook share button and try again. Copying the URL directly into Facebook to share something works fine now!