I think the key issue that you are having is summarized in this statement:
...but most every document in question is Adobe CS.
But first, let's make sure you have a foundation on permissions and the setup of the share. In your note, you mention that there are three groups that have access to the share. Does your share look something like this:

Note that I've color highlighted the permissions to show the Posix permissions and the ACL.
In red, starting from the bottom and working my way up, Everyone is read only, the default group (wheel) is read only, the owner (carbon) is read and write. When viewing from Terminal, this would be the equivalent of using the ls -l command to list the contents in long form:
drwxr-xr-x+ 2 carbon wheel 68 Mar 25 14:55 SampleShare
Starting from the left, the d means this is a directory. The next 9 characters can be broken into three columns, one for the owner (rwx), one for the group (r-x), and the final for everyone (r-x). Note the + at the end. That signifies that the folder also has an ACL applied.
Using ls -le will reveal the ACL and show a complete view of the permissions for the parent folder (this is the folder that is your shared folder):
drwxr-xr-x+ 2 carbon wheel 68 Mar 25 14:55 SampleShare
0: user:_spotlight inherited allow list,search,file_inherit,directory_inherit
1: group:group1 allow list,add_file,search,add_subdirectory,delete_child,readattr,writeattr,readextattr,writeextattr,readsecurity,file_inherit,directory_inherit
2: group:group2 allow list,search,readattr,readextattr,readsecurity,file_inherit,directory_inherit
3: group:group3 allow list,search,readattr,readextattr,readsecurity,file_inherit,directory_inherit
In addition to seeing the POSIX permissions, I also can see the ACL that contains the three groups defined when setting up the share (group1, group2, and group3). Group 1 has the greatest of rights, being able to read and write while group 2 and 3 only have read access.
Once this is set on the parent folder, child folder and descendants will inherit the ACL permissions from the parent. This is represented by the following attribute within each ACE:
file_inherit,directory_inherit
However, and I believe this is the key issue that you are experiencing because of the age of your Adobe stack, POSIX permissions do not support inheritance. It was removed in 10.4 or 10.5 (sorry, memory of those versions is getting a little fuzzy). That means that if you do the following on your parent share:

Note the change was to set the Group 1 (group with Read and Write access in the ACL to the primary POSIX group (POSIX can only support one group)) and set the permissions to read and write, then my view of the share is now:
drwxrwxr-x+ 2 carbon group1 68 Mar 25 14:55 SampleShare
0: user:_spotlight inherited allow list,search,file_inherit,directory_inherit
1: group:group1 allow list,add_file,search,add_subdirectory,delete_child,readattr,writeattr,readextattr,writeextattr,readsecurity,file_inherit,directory_inherit
2: group:group2 allow list,search,readattr,readextattr,readsecurity,file_inherit,directory_inherit
3: group:group3 allow list,search,readattr,readextattr,readsecurity,file_inherit,directory_inherit
Note the POSIX is now drwxrwxr-x and the primary group is listed as group1. This all looks great on the parent folder.
Ah, but now I log in as a user to the share and I create a file and a folder at the root level. I've logged in with a user who is a member of Group 1 so I can create this content (members of groups 2 and 3 can only read based on the ACL). Looking at the permission of the file and the folder I find: