Hello, if the sites were actually shut down, then they must have been causing an issue with the server. Please understand, there are a lot of other people on a shared server and we have to protect all their sites as well as the server from crashing when one account causes issues.
I can certainly understand the slow connection issue and you do have a lot of files. It actually is a very large amount. I would not be able to help move them because of my remote connection but I do have a suggestion. On the old server, run a full backup under Backup in Cpanel. Set that backup to FTP the backup file to your new server to the root folder. Once you get the file over there, then you can extract files, move them to where they need to be, setup databases, etc or someone here in the forums can help. Just need to get the file over there first.
I would also go through them and look for any files you did not create or upload to be sure the old account was not exploited thus causing the issue.
Thank you, i should
Backup Destination: Remote ftp server ?
Remote Server (FTP/SCP only):
Remote User (FTP/SCP only):
Remote Password (FTP/SCP only):
Port (FTP/SCP only)
Remote Dir (FTP/SCP only):
Could you kindly help me out with this by fillin in, My remote directory that i am uploading files to how will the direction be line? var/

?