As a Firefox on Linux user I checked one of those sites that tries to estimate how many bits each public aspect of your setup reveals about you. It turned out available fonts was by far the most unique aspect of my setup.
The only surefire way is to disable javascript, extensions, cookies, etc. https://browserleaks.com has a pretty good breakdown of the different techniques you can use. There's another JS technique that probes the hardware to fingerprint a browser too.
Use Tor Browser even if your not using Tor if you're looking for better privacy. It's modified to mitigate as much as possible. Facebook is just bad. Avoid it at all costs if you value privacy. And it's not just facebook. Sites like facebook, google, etc also use several 3rd party "advertising" (i.e. data gathering) companies to gather data and build profiles on users and share that data with each other. Even on your regular use browser I would highly recommend uBlock Origin and Privacy Badger.
But with such a unique browsing situation you're basically identifiable on that basis alone. Your best bet would be to have your browser present itself as a common browser on a common platform, and block tracking and ads.
UserAgent is still top culprit (16 bits of identifying information) followed by browser plugins (12bits) then WebGl (12b), canvas (9b), language (if not english nor chinese) and then fonts at 5bits.
One thing I don't really like about that site is that it gives browsers worse scores for not unblocking third parties which promise to honor do not track. Surely you're more safe when you don't trust anyone instead of trusting that third parties which honor DNT actually honor it. It kind of reeks of pushing an agenda, which would have been okay (it's the EFF after all) if the tool didn't claim to score your browser on how well it protects you from tracking.
http://yinzhicao.org/TrackingFree/crossbrowsertracking_NDSS1...
Use Tor Browser even if your not using Tor if you're looking for better privacy. It's modified to mitigate as much as possible. Facebook is just bad. Avoid it at all costs if you value privacy. And it's not just facebook. Sites like facebook, google, etc also use several 3rd party "advertising" (i.e. data gathering) companies to gather data and build profiles on users and share that data with each other. Even on your regular use browser I would highly recommend uBlock Origin and Privacy Badger.
https://github.com/gorhill/uBlock
https://www.eff.org/privacybadger