Need help moving a site to new cPanel based host
PermalinkUpdate: I believe that the site is running version 5.6.1.2

I can do this asap if you like shouldn't take that long.
You do realise that you must change the dns settings to point to the new server and allow up to 24 hours for those settings to propagate globally before the site will 'appear' on the new server?
Normally this would be a simple task for a dev that has the knowledge and experience required.
I am having 6+ Years of Experience in building Concrete5 Websites. I can help you out straight away. Kindly lemme know the requirements..
I can help you out in Migrating your website to new server and will take a few minutes.
I can start straight away..Lemme know.
Thanks.