If you are trying to get backups of your sql databases you should be able to do this through phpMyAdmin, which should be available with your godaddy hosting. This would just give you a .sql file to download directly from your browser and shouldn't produce any similar issues.
As far as resuming transfer, as long as godaddy's server for your hosting supports it (which I see no reason why it wouldn't) it should be automatic and entirely possible in filezilla. Unless the transfers themselves aren't registering as failed transfers, in which case you'd be out of luck. But back to filezilla... if you open up the 'Settings...' under the 'Edit' menu option you can click on 'File exists action' located underneath the 'Transfers' branch. Here you can set the default to 'Resume file transfer'. This should automatically resume your file from where it left off when you start the download again after your connection times out. It actually should have asked you about this when you reconnected and you could have selected this option but... for now let's just see how things go and if it solves your problem, great.