You might have a couple of different issues going on here.
Hopefully my notes below will help.
The only issue I see is that, when I did the "Publish", I got a number of errors that say: "Notice: A non well formed numeric value encountered".
This is a message which appears when running Showkase v1.7.0 (or earlier) under PHP 7.1 (or later).
It sounds like your new web server may be running PHP 7.1 or 7.2 (and this is why you are seeing this message for the first time with your Showkase v1.6.1 site).
Showkase was updated to be compatible with PHP 7.1 and 7.2 in Showkase v1.7.2.
I checked the site System/Setup menu under showkase and it has the correct path.
Did you actually try running Setup (rather than just checking the path on the Setup page)?
Running Setup is a crucial step when moving or renaming your Showkase installation (as noted here).
I would try upgrading your Showkase site (full instructions here) to the current version (v1.7.3) and running Setup.
If this does not work, then it might be quicker to recreate the site.
If you create a fresh Showkase installation on your web server, you can then:
(1) Retain the Showkase settings from your original site by manually copying the following files and folders from your original site to your new site (overwriting the default versions which will be in place after a fresh install).
_data/sitedata (folder)
_data/themedata (folder)
_data/pagesprefs.txt (file)
_data/theme.txt (file)
Do not copy the _data/pagesdata.txt file though, as this lists the pages present in each site (which will be different after a fresh install).
Also, if you use a custom logo via an image stored in the library, for example, then you might also like to copy the '_library' folder from your original site to your new site.
(2) Import all the pages from your original site (via 'Site -> Import') to save having to recreate all your pages from scratch. You can import multiple pages at once so recreating your original site should be relatively quick and easy.
Hopefully upgrading Showkase and running Setup will work for you but the procedure above should also work fine as a backup plan.
Just be sure to create a full backup of your site before you try anything (just in case anything goes wrong and you need to reinstate your original files at a later date).