> I run a very basic VPS with 256MB of RAM.
That is extremely tiny by today's standards. Use the link that Peter gave you to compute the buffer sizes, then cut them in half. After that, monitor swapping. If your tiny RAM is swapping, you will need to decrease things further.
Set query_cache_type=OFF.
Turn on the slowlog; see what shows up in it; then we can discuss specific queries. Granted, this does not address your original question about the upgrade, but it may lead to better performance.
If you do these after being up a few hours, and present the outputs here, I can dig further:
SHOW VARIABLES;
SHOW GLOBAL STATUS;
I can't judge the buffer settings without knowing whether you are using MyISAM or InnoDB. Note that the default has change from MyISAM to InnoDB; this _may_ have affected your tables. If so, we should look at SHOW CREATE TABLE together with the details in the script to see if something is worse because of the engine change. Further tips in that direction:
http://mysql.rjweb.org/doc.php/myisam2innodb