This might be a little difficult to troubleshoot, especially as there's no error message to point us in the right direction.
I don't think the fact that the Contact Form used to work from a different folder is part of the probleem. I can't think of any reason why the Contact Form should work from one folder but not another. (Thank you for letting me know, though. The more information we have about a problem, the better.)
However, the fact that it used to work (albeit in a different location) but now does not suggests that something may have changed on your web server (round about the time you moved your Showkase site) to have caused the problem.
Try asking your web host to see if they have changed anything on your hosting account recently. Perhaps they've changed the PHP version and the new version uses different settings than the previous version.
Also, if you have access to your web server's error logs, please take a look at them. There may be an entry corresponding to each non-functional click of the 'Send' button which might give us a clue as to what is going on.
I expect you may already be using the latest version of Showkase (v1.7.6.) but, if not, please try updating as the internal version of PHPMailer was updated in Showkase v1.7.6. If necessary, instructions for Upgrading Showkase can be found here.
It's a long shot (and it shouldn't really be necessary) but you could try using the very latest version of PHPMailer (v6.6.4).
I've uploaded the file you'll need here: [Link removed.]
I've not zipped the file so when you click on the link above, you'll see the file's code. Just click the Download button near the top right of the page to download the file to your computer's hard drive.
Now, the file you'll need to replace within your Showkase site is showkase/admin/plugins/contact/master/contactcore/PHPMailer.php
Make a backup of the existing 'PHPMailer.php' file first, though (just rename it before uploading the new version), in case you need to reinstate it at a later date. (Remember this new version has not been tested by the Showkase developers.)
After replacing the 'PHPMailer.php' file, you'll need to republish your site (just click the 'Publish' button) for the new file to propagate throughout your site.
I hope this helps or at least points you in the right direction.