You either use TOR (only way to browse without an unique fingerprint) or have a hardened Firefox without using most sites that could potentially track you.
Browserleaks is not extensive but it gives you a good idea of how to reduce that fingerprint https://browserleaks.com . Another option is to use https://github.com/arkenfox/user.js but be sure to read their wiki
Having a separate partition is a good idea and funny enough it can help you in this issue. If you try to reinstall the system, you can format the other partitions and leave /home untouched, so you will have a fresh system but keeping all your personal data as it is; that’s the main benefit of having multiple partitions.
Next time, you should just make /var bigger according to your own experience. Other people only have 2 partitions: / as a 20-40 GB one (or bigger, this is according your needs) and the rest of your HDD/SDD for /home. Another tip is that when you’re re-installing the system, use a new nickname, otherwise the new system will try to use all the old config files in your old account, and finally, if there’s very important stuff in /home, definitely make an outside backup, there’s always the chance of misreading something or ticking the wrong box.
Music player, fork of the now defunct Clementine. Lots of options to play around.