I am trying to update my production and shipping times, and no matter what I do it's not showing an update when I go to the listings page in my shop.
You must edit your shipping profile - did you do that?
Yes, that's what I've been doing. I've been here 10 years and I don't know why different items are showing different dates when they are all under one shipping profile.
Clear your cache and see what happens. Try a different browser. Sometimes changes don’t show right away. You can always recreate the shipping profile and assign it to the listings … delete the old profile … that usually knocks out any glitches.