This is a major issue for our storefront as well. I have been back and forth with Amazon support on this too. Also, bounced around as one would expect. Here is what I am finding, some info from Amazon support, some from effin around and finding out:

It seems that when the new Seller Center was rolled out, the global item specs that gave our shipping templates the length x width x height and weight of the items were switched from working for all sellers to just Amazon themselves. So if you are on a listing that had that information on it already, you likely need to go into your own listing and add these numbers if you had not before. We have tied most of our items to existing listings and so relied on those numbers that were already there.

In addition, it seems there is an issue with the "Office Supplies" category specifically. Adding dimensions and weight to items in that category, ensuring the proper shipping template and even testing these items with my personal amazon account, I would see the correct fee and shipping rate being charged yet our store still received orders with no shipping fees. Even tested a price change and orders came in after the change in price at the previous price like it didn't recognize that we changed the price, and still with free shipping. That seems to be category specific though since the changes made to other items seemed to take affect.