I have been doing some testing and I get the cookies to show in the bindings regardless of how I add them. As a test please try this, add a new cookie from the server behaviors, WA Cookies Toolkit > Set Cookie Value. See if this adds a bindings to the list for you.
If not there may be some sort of corruption that is occurring for you. Have you tried this on more than one page or in more than one site? It may be related to the site or page you are working with and testing on another may work.
If this does not work there may be an issue with the cache file of some type. You can clear out this cache file to have DW recreate it properly. Here is a link to a thread that list the locations to find this cache file and delete it.
Post 6 has the locations.