Have you spoken to HostGator about this yet?
As I understand it, and please let me know if this is not correct, shared servers allocate a space allottment to a user, you cannot go over that allottment, you try and you get errors - lots of them. Could this be a cause? Have you checked your server logs? What do they say?
One way around these kinds of issues is that when you upgrade your Moodle, you do it over the top of your existing Moodle, that is, you delete all your Moodle files, not the database, not the moodledata folder and definitely NOT the config.php file, just the moodle php files. You then unzip the newer version over the top of the now blank Moodle. Log in as per normal and it should begin the upgrade on its own. I have done this from a 1.9.13 to a 2.2.3 and it worked, but it did not have that many users, 112 right now, and less than 60 courses. Worked well actually, much to my surprise, but that is not a shared server...