You'll need to change your IP as well. Something you have never logged in as to create the account. Sock puppet accounts can be associated by IP and social graph searches.
I used VPNs. But FB keeps a track of a lot of public VPN & Cloud providers and then throws a ton more "captchas" your way - asking for your picture, govt id, phone number verification etc.
As for searches - I've searched a lot of random stuff totally unrelated to my personal account JUST to throw FB off while acting like a real user (liking, reading, scrolling etc.).
It only takes one to five high probability data points to link you to the original identity near flawlessly. Something rare. Clicking on a nonpublic person is one of those.
