This is a very shallow way of "escaping"; your browsing activity is definitely being linked to your social media identities via your IP(s) and browser fingerprint.
For instance Google does maintain such non-cookie-based user identities. I'd be highly surprised if Facebook didn't as well; your data is just too valuable to pass on such easy fixes.
The browser fingerprint isn't going to help them though.
Of course the only real solution (other than cutting yourself off completely) is for there to be a stable, secure, reliable, non-tracked, ad-free alternative that enough people use - and that isn't going to happen unless you have a spare high improbability generator handy.
For instance Google does maintain such non-cookie-based user identities. I'd be highly surprised if Facebook didn't as well; your data is just too valuable to pass on such easy fixes.