Your host should be able to assist with this.
Basically, if you've backed up the entire account using the FULL backup tool in your cPanel and have downloaded it, you should be able to kill everything in the public_html directory safely. The backup is generated to the root of the account, outside public_html so you should kill that backup as well.
You might want to wait till you've got the new site up and running perfectly before killing the old one though, just as a precaution.
Once you've moved the domain name to point to the new account on another server, the old account is worthless and can't be accessed by anyone but the owner of the server. He'll wipe the account anyway which would do the same thing as you'd like to do here.
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