What I mean is that I would first create a new database (using cPanel, but there are other ways), which creates essentially an empty bucket, make sure that collation is correct, then issue the import command from an SSH session. This process brings the data (and tables) from the old to the new. Your method might also work (in theory), but I too found Moodle and the Linux server environment "fussy." I remember trying all kinds of ways to do this, but when one method finally worked, I documented it and continue to use it.
Do you know how to use SSH?
I keep thinking about other differences that you have mentioned. For example, there could be something going on with different versions of mySQL. You also mention having add-ins in your current moodle (which if not loading correctly, could be causing your problems). And are you installing the exact same version of Moodle?
Unfortunately, I am not a linux expert so I too find some of this a bit of a mystery. You have Apache to work with, mySQL, php, and Moodle. In theory, they should all work well together, but they can overwhelm a beginner (and an advanced beginner like myself). But I have to tell you that when you get Moodle running, it runs like a charm. I am not trying to paint a grim picture, Moodle has performed extremely well for me for six years (and so has goDaddy), but I have had to learn many things (which I actually enjoy.)
Have you tried reinstalling a copy of your backup to the exact same server (different database and directories) to verify that your backup is good?