If you check the post on the backend, what is the url of the featured image as if its not the URL of the current domain then that is what our shortcode is pulling.
Hi, Shane is on public holiday so I'll try to help. If you go to Toolset > Settings > Custom Content, you will see a couple of options for clearing the image cache - try both. When you migrate a site, you may need to clear the Toolset image cache to update those URLs correctly. Please try this the next time you migrate a site, and let me know if the issue is not resolved.
I downloaded a copy of this site to see what's happening, and it appears your find and replace script is missing the guid column of the posts table for the posts shown in this archive, among other things. Check wpstg0_posts, ID 1136. The guid of this post is pointing to the old domain, which is the main problem here. As soon as I replace that domain with my local test environment domain, the problem is solved.
I checked the SQL dump file generated for my clone and there are 1084 instances of the old domain showing up, so I suspect your find-and-replace script is not set up to handle all the necessary tables, columns, and serialized data structures, or perhaps it's not flexible enough to handle minor URL differences like http vs https. Is there a reason not to use Duplicator or some other migration tool instead of the find-and-replace script? It might save a lot of headache.