You should be able to do this.
1. I always check to see if links should be done and do those first, before transcribing or correcting. Make the links and save. Then I usually open one of them in the OLD transcription window, as it is easier to see all the photos and the individuals at the same time.
2. For each person, open and edit and then save. I go down through each one this way, and after all are saved in this screen, hit reload/refresh for that page and check that the corrections are actually there.
3. For some reason, just now since the latest upgrades, the Save function does not replace the old data with the new in what you are viewing. It is actually replacing it, but you can't see it unless you reload/refresh. Not sure why but that is what I am seeing now.
If you only have one or two images to link, this work in the NEW Screen but I was recently doing corrections where there were 20 or more multiple images and duplicate entries, every one of which needed corrections, and I found that you can't save one, without going through every single image in the transcription set. That is OK as it helps not to miss an images, but the OLD screen works much better for multiple image links, like we often have in large family plots.