Sorry to say but you are doing many things wrong.
To start off, if you are using Nginx you don't need Cpanel/WHM/Plesk offered by Hostgator.
You are already spending $200/month on hosting (ref)
So better get a dedicated server without bloated WHM/Cpanel.
WordPress needs PHP, MySQL & Postfix (a sendmail program). You can install them from easily using this tutorial.
Next, wordpress needs RAM. You have 4GB for $200. Again you can get much more RAM per $ using http://servercraft.co/dedicated-servers/ or some other dedicated server provider.
Now if you are happy with Hostgator (which gets dedicated server from SoftLayer by the way), rather than using a DB manager plugin, you can use Hostgator control panel's to backup mysql.
Or from command line, schedule a mysqldump.
If you want to otpmize mysql, you can do using a another cron job
Finally, mysql needs tweaking. It you don't do it nothing will help. Please refer this to tune your mysql - http://rtcamp.com/tutorials/mysql-optimization-mysqltuner/