I went through the same problems after insert into post but this solved the issue by accident.
Once you “inserted into Post” it will take you back to the Sponsors Carousel Settings Page. Look at the “New (select *full size* before insert)” field it should contain the Image URL that you just added.
Click the field, hit Enter.
The image instantly appears below. As you add images, there are up and down arrows to change the order of image links.
Another Tip if you want to add a Website URL, type it into the Image “Caption” field on each image. It seems easier to do when you first upload the image, but if it’s existing you can use the media library to edit images at anytime.
Last Tip to make the sponsor logos show up, I cut and pasted the <?php echo sponsors_carousel(); ?> into the footer.php file. It can be found in the dashboard under appearances, editor. I added it just before </body></html> and it sits right at the bottom of the homepage.