Your current website host should also have provided you with FTP access, the login may be the same as what you use to get into the website builder, or you may be able to set it to something specific in the control panel. I have just had a look at your site, it looks good but beneath the skin there is a lot of extraneous code that has been stuffed in there by the website building software. You've got an extensive site... it would be a lot of work to reconstruct everything. If you can get some cooperation from the current host's helpdesk, get them to figure out how to allow FTP access and use that to download the complete contents of the public_html folder (or whatever it is named on their system, the helpdesk should know). Getting the system to e-mail you a zip file most likely fails because the resulting file is way too large to fit in an e-mail.
The Wayback archive can't help you here, only as a reference as to 'this is how it should look' as
netstruggler mentioned. Ideally, you should rebuild the site with another website builder (or something like Dreamweaver or equivalent) so that all the extra code is removed as it only serves to slow things down and keep Google in business. But that's just my view.