Thank you both for so quick replies.
Disabling plugins or theme wouldn't help, everything was working fine on a testing server, and the 'get_permalink' code just stopped working after moving site to the new server and new domain. The way the URLs are dispalyed now is:
http://www.newdomain.com/<?php echo get_permalink(17); ?>
(but only those were I used get_permalink in the template, all other links work absolutely fine)
All old URL instances in a database has been replaced (I used Duplicator plugin).
After moving the site I resaved the permalinks in Settings, would that fall into what Timtohy has suggested? I do not have access to .htaccess file using ftp, although my ftp is showing hidden files, I do not have access to control panel (client's server), and host is saying the file is visible, but I still can't see it, even when they changed permissions to 777
All friendly urls and links are working on the site, just apart those put manually in the template using get_permalink code